Note: In my opinion, this is a complicated subject which I have done my best to condense as much as possible, but nonetheless I apologize for it being so long. There's a tl;dr at the end, if you prefer that.

As I pointed out in my comments on @LangLangC's answer, the problem with both the question and that answer is that they're based on the assumption that the Rabbi Yishmael in the midrash, whose full name was Rabbi Yishmael ben Elisha, lived in the time of Hadrian. However, this is heavily disputed and the true identity of Rabbi Yishmael ben Elisha is the subject of much debate. Therefore, to dispute the historicity of the source in question simply by stating that Hadrian did not have a daughter, which is what the questioner (or their friend) alludes to and what @LangLangC wrote explicitly, is not a very strong argument. I'll try to outline the general frame of the debate as best and as simply as possible to illustrate the issue:

In BT [Babylonian Talmud] Brachot 7a it says:

"It was taught in a baraita that Rabbi Yishmael ben Elisha, the High Priest, said: Once, on Yom Kippur, I entered the innermost sanctum, the Holy of Holies, to offer incense, and in a vision I saw Akatriel Y-a [one of the names of God], the Lord of Hosts, seated upon a high and exalted throne..."

Furthermore, a descendant of his, also named Rabbi Yishmael stated in Tosefta Challah 1:10:

"Said [Rabbi Yishmael] to them: "I vow that by the garment that wore my father and by the tzitz that wore my father..."

So here we see that RYbE was a High Priest. However, Josephus gives the names of the High Priests of both the First Temple and the Second Temple in Antiquities and does not mention RYbE. Now, an argument can be made that when RYbE is called a "High Priest" (כהן גדול), it is merely a title that is intended to depict his prominence, as was the case by many important priests from circa the Second Temple era who were not all literal "high priests" (see for example Menachem Stern's Hebrew essay "The Politics of Herod and Jewish Society towards the End of the Second Commonwealth", p. 249 and n. 92 there). However, one would then have to contend with the term "נכנסתי לפני ולפנים" (lit. "I entered before and inside"), as entering לפני ולפנים (before and inside) is a term typically used to refer to entering the Holy of Holies, something only done by the high priest on Yom Kippur. In this case, an argument might be made that the baraita here is borrowing the term to express a kind of deep kabbalistic mental state that RYbE entered. This is certainly a possible argument, though the choice of terminology would certainly be highly irregular and would demand an explanation of its own. And then there's still Tosefta Challah to be explained.

However, an argument can also be made against Josephus. It is well-known that Josephus did not include every bit of information and also more than occasionally deviated from the truth. One of the most famous examples is his attempt to defend Titus with regards to whose idea it was to destroy the Temple at the end of the Great Revolt: Josehpus (Wars VI:4:3) claims that Titus was against the idea while his generals supported it and eventually he allowed his men to burn the Temple in the heat of the moment (pun unintended) (ibid. 5-7). Tacitus (Fragments, 2) on the other hand, states that it was Titus who fully supported the idea and his advisors were the ones that opposed it. Remember that Tacitus was no lover of the Jews (see for example here) nor did he have any problem in depicting the Romans enacting their full strength. Josephus, on the other hand, had to both present the Romans in a positive light and to somehow present their actions as necessary evils at worst or simple mistakes at best.

So Josephus may not be 100% trustworthy on various matters.¹ It is therefore entirely possible that Josephus simply chose to not make mention of RYbE in his works. I am aware of two reasons why this may be so:

1. As suggested by Rabbi Tuvya Tavyumi in his essay in Hebrew "Flavius Josephus", Josephus was not a Pharisee but a kind of Hellenistic Jew whose ideology was similar to that of the Hellenistic Jews of Alexandria. As such, he was vehemently opposed to the Jewish sages. For this reason, he hardly ever mentions sages by name, not even some of the most prominent ones, such as Hillel (and another matter of debate is whether Sameias is Shammai. See for example Zeitlin's essay for one perspective). This would explain why he would have excluded the name of RYbE. When he does name sages, he seems to be consistent in that he either names them when he is copying verbatim from older sources (such as Nicholas of Damascus) or that he personally knew them and had close ties to them, as is the case with Rabban Shimon ben Gamliel I or Rabbi Yehoshua ben Gamla. Anyone else got the boot.

2. He was attempting to protect the family of RYbE and other sages and their families from the wrath of the Romans. This may be proved by how Josephus mentions that the Prince of Judea, Rabban Shimon ben Gamliel I, was from a prominent family (The Life of Flavius Josephus, 38) but doesn't explicitly state what makes their genealogy special: That they were descendants of the House of David (JT Taanit 4:2 and hinted at in JT Ketubot 12:3, JT Kila'im 9:3 and Beresheet Rabbah 33:3).² Indeed, Eusebius in Church History 3:12 records that after the destruction of the Temple, Vespasian hunted down the descendants of the House of David. Perhaps Josephus purposefully excluded as much information about the sages as possible to protect them (consider that by the time that appendix was written, Domitian, who greatly hated Jews, was already in power).

So far I have presented views that attempt to call into question the legitimacy or at least the simple understanding of different Jewish sources from antiquity. But there's another view, and that is that Rabbi Yishmael ben Elisha is actually mentioned also by Josephus, but by a slightly different name, as Yishmael ben Piavi (or Fabi). This man is mentioned a number of times in Antiquities, and two times in particular when he was appointed as high priest: Once in XVIII:2:2 and once in XX:8:8. Some scholars believe that these sections refer to two different, but related individuals, while other think that this refers to the same person, appointed twice to the role of the high priest. Yishmael ben Piavi (the second?) was apparently eventually put to death in Cyrene (Wars VI:2:2).

This name appears also in Tannaitic and Amoraic literature. According to the Tosefta Menachot 13:4 (mentioned also in the Talmud), "Piavi" or "Yishmael ben Piavi" was actually the name of one of the houses of the high priests. When Yishmael ben Piavi is referred to as "ben Piavi", it doesn't necessarily mean that his father's name was Piavi - it may simply be a title referring to his house. Another possibility is that Piavi simply had another name, Elisha (which may explain the Tosefta's mentioning of a "House of Elisha" - perhaps a branch of Piavi). See for example Yitzchak Baer's essay in Hebrew "The Service of Sacrifice in Second Temple Times", p. 136 for some possibilities on how this might work out. This Yishmael was in fact a sage - some sources refer to him as Rabbi Yishmael ben Piavi (Mishna Sotah 9:15, BT Yoma 35b, JT Sotah 9:16).

The long and short of all three of these positions is that Rabbi Yishmael ben Elisha lived circa the end of the Second Temple era, in which case, it is actually possible that he was killed by Casear's daughter - Domitila. Now, Domitila died before Vespasian became emperor, but then again, we also don't know when Yishmael ben Piavi was killed exactly. It might have been right at the start of the war. Interestingly, Titus first came to Judea with a legion in the year 66 CE from Alexandria, which is fairly close to Cyrene. Of course, one could still argue that this would mean that RYbE died before Rabban Shimon ben Gamliel I.

Another possibility is that the daughter in question is the daughter of one of the previous emperors. From what I can tell, the only relevant candidate is Vitellius's daughter Vitellia. Not only was she a daughter of an emperor (albeit, a short-lived one), she was also close to Vespasian, who succeeded her father.

Now the other side of the debate: There's also the possibility that Rabbi Yishmael ben Elisha lived during Hadrian's time. It has been suggested by some scholars over the years that the Jews during the time of the Bar Kochba Revolt managed to not only conquer Jerusalem but to also set restart the Temple service in some capacity (see for example Dr. Yehoshua Peleg's essay in Hebrew "The Time of the Temple in Tractate Middot", Techumin 18, pp. 480-498). Of those hold this view, usually it is thought the name "Elazar Hakohen" that appears on many of the Revolt's coins (for example) was the high priest of this Temple, but it is also possible to suggest that he served in a different capacity in the Revolt and the high priest was actually Rabbi Yishmael. The problem with this view is that Rabban Shimon ben Gamliel II was not killed during or shortly after the Revolt. One could then put the apparent relationship between the two down to either artistic license, or to the possibility that a younger Rabbi Yishmael had actually known Rabban Shimon ben Gamliel I, some sixty years prior (in some variance, this was suggested by Rav Nissim Gaon), or that the "Rabbi Shimon" here isn't RSbG but another Rabbi Shimon (some have suggested Shimon ben Azzai). Another problem, which is the main issue of the OP, is the matter of Hadrian's daughter. It is possible to suggest that as Hadrian's sons were not really his own, so too this daughter was not really his but refers, instead, to a woman who was close to him, such as Julia Balbilla, as may be suggested in the case of Vespasian.

There are a number of various other views as well, but I think these are the main ones. I hope I have succeeded in explaining why simply showing that Hadrian didn't have a daughter isn't enough to invalidate the whole story. I think much more research is necessary for that, and even then - I don't think there's any definitive answer.

As a side-note, it is worth noting that the Talmud states in at least two places (from what I found) that the scalp of Rabbi Yishmael was preserved for a number of generations by the Romans (see BT Avodah Zara 11b and BT Chullin 123a). These are non-midrashic accounts which lend credence to the validity of the frame of the story at the very least.

TL;DR: The gist of all of this is that while yes, Hadrian did not have a biological daughter, that in itself is not a strong enough case to question the historicity of the story.

¹ Anecdotally, I have heard from one of his students that the late Professor Moshe Ber used to open his "Introduction to Jewish History in the Second Temple and the Mishnaic and Talmudic Eras" university course by announcing that "everything Josephus says is a complete lie, but unfortunately he's all we have" and would then proceed to base much of the course on the study of Josephus...

² On the validity of this genealogy, see Rabbi Dr. Reuven Margolies's short note "On the Genealogy of Rabbi Yehudah the Prince", Sinai 39, pp. 104-105.

Straight short answer to the question as phrased in title and repeated in the question body:

No.
Hadrian had no children of his own, and thus no daughter. He also did not adopt any daughters.

From all clues considered present in the story, the emperor itself was also not even present when the executions took place, making it unlikely for his (fictional) daughter to interfere in any way. The text giving rise to the assumed reading 'must talk about Hadrian's daughter' is not a reliable historical source, but religious literature, based on some actual history, but conflating all the 'true information' — from 'time' over 'place' to 'names/identities' and more — into one narrative that deviates from the historical record.

On how to read the question

There is this text called Eleh Ezkerah which tells the story of ten men executed seemingly in very short succession under 'a Roman emperor', which from context insinuates that it must be Hadrian who the emperor/caesar refers to. Given that the text mentions 'the emperor's daughter', it is legitimate to question this daughter's very existence.

Q So which one is true?

Means in my reading of the question: one historical source with a strong religious background claims that an actual emperor's daughter was being present and interfering with an execution of a historical figure, but most other historical research excludes the very possibility of at least this part of the story to be true, as the emperor mentioned in the story had no daughter at all. So, how do we weigh the arguments?

General reliability as a historical source of Eleh Ezkerah

This religious legend is unreliable as a historical source in numerous ways. Hadrian had no daughter, Hadrian himself was not anywhere near when the Rabbi Yishmael was killed, Yishmael was killed even well before Hadrian became emperor; that is: Yishmael was killed before the bulk of the Ten Martyrs anyway. The Eleh Ezkerah is a literary source, based on some actual history, but not a straightforward historical document giving reliable historical information, but rather fictional in just too many of its details.

The biographers of Hadrian make it explicit that Hadrian remained childless

The text giving rise to this question is quite ahistorical in a few aspects. Hadrian not only 'was gay', but also 'properly married', with Sabina, but these historical possibilities seem to lead to a quickly found dead end, answering the title of this question in a short summary:

In the meantime Hadrian had experienced a personal trauma. He had been married at the age of twenty-four to a distant kinswoman, Sabina, a grand-niece of Trajan. The marriage was childless and – at least after two decades – loveless. Hadrian was in any case more interested in males.

— Anthony R. Birley: "Hadrian. The restless emperor", Routledge: London, New York, 1997.

Not every aspect of Hadrian's life can be uncovered, but in the narrative of the Eleh Ezkerah story the executions took place on one day and center around the time when Hadrian was emperor.

The clues to be read from Eleh Ezkerah

The clues from Eleh Ezkerah form irreconcilable conflicts and contradictions to established history, internally from the story itself and externally relating to other sources.

Concerning the text Midrash Eleh Ezkerah originating this story

There are conflicting accounts of his martyrdom. The Avot of Rabbi Natan states that he and Shimon ben Gamliel were decapitated in quick succession. However, the Midrash Eleh Ezkerah relates that the Caesar's daughter was so taken by Ishmael's beauty that she ordered his head be skinned while he was still alive so that she could stuff it with straw and preserve it, and this is how Ishmael died. The Eleh Ezkerah is, however, known for its use of poetic license at the expense of historical accuracy, and its author(s) probably changed the narrative in order to produce a greater effect upon the mind of the reader.

— Wikipedia: Ishmael ben Elisha ha-Kohen

The life-times of The Ten Martyrs alone should suffice to demonstrate that those people did not all die as described in the text—at the same time, nor even all when Hadrian was in charge. One of them, Simeon ben Gamliel, is for example known to have died in 70 AD, while Judah ben Dama died 136 AD, while Rabbi Akiva was dead by 135 AD. Akiva's successor Eleazar ben Shammua however outlived them all and grew quite old.

To quote Wikipedia again for the 'martydom of Yishmael':

There are conflicting accounts of his martyrdom. The Avot of Rabbi Natan states that he and Shimon ben Gamliel were decapitated in quick succession. (link) a dramatic poem (known as Eleh Ezkera) tells their story as if they were killed together. […] The Eleh Ezkerah is, however, known for its use of poetic license at the expense of historical accuracy, and its author(s) probably changed the narrative in order to produce a greater effect upon the mind of the reader.

The assumption that anything relating from that part of the Eleh Ezkerah about 'killing Yishmael and any daughter intervening' has to relate to 'Hadrian's daughter' is not very viable. According to Jewish Encyclopedia as well, the Ten Martyrs:

Among the numerous victims of the persecutions of Hadrian, tradition names ten great teachers who suffered martyrdom for having, in defiance of an edict of the Roman emperor, instructed their pupils in the Law. They are referred to in haggadic literature as the 'Asarah Haruge Malkut. Popular imagination seized upon this episode in Jewish history and embellished it with various legends relating the virtues of the martyrs and the fortitude shown by them during their execution. These legends became in the geonic period the subject of a special midrash—the Midrash 'Asarah Haruge Malkut, or Midrash Eleh Ezkerah, of which there exist four versions, each differing from the others in various points of detail (see Jellinek, "B. H." i. 64, vi, 19). Contrary to the accounts given in the Talmud and in Midrash Rab bah ('Ab. Zarah 17b, 18a; Ber. 61b; Sanh. 14a; Lam. R. ii. 2; Prov. R. i. 13), which clearly state that there were intervals between the executionsof the ten teachers, the Midrash 'Asarah Haruge Malkut, probably in order to produce a greater effect upon the mind of the reader, describes their martyrdom as occurring on the same day.

Their Names.

This midrash differs from the older sources in regard also to the accusation upon which they were condemned. It says that when a certain Roman emperor who had been instructed in the Law came to the Biblical passage, "And he that stealeth a man, and selleth him, or if he be found in his hand, he shall surely be put to death" (Ex. xxi. 16), he conceived the following mischievous device: He summoned Ishmael ben Elisha (perhaps the propounder of the "Thirteen Rules"; see Ab. R. N., ed. Schechter, p. 54b; comp. Ned. ix. 10), Simeon (certainly not Simeon ben Gamaliel II; see Ta'an. 29b), Ishmael, Akiba ben Joseph, Hananiah ben Teradion, Ḥuẓpit (the interpreter ["meturgeman"] of the Sanhedrin of Jamnia), Yeshebab (the secretary of the Sanhedrin), Eliezer ben Shammua' (in Midr. R. l.c. "R. Eliezer Ḥersanah," or "R. Tryphon"), Hananiah ben Ḥakinai (in Midr. R. l.c. "Judah ha-Neḥetam"),and Judah ben Baba, and demanded of them what was the punishment prescribed by the Law for stealing a man. They answered, "Death"; whereupon the emperor said, "Then prepare to die, for your ancestors [alluding to the history of Joseph and his brethren] committed such a crime, and you, as the representatives of the Jewish nation, must answer for it."

The biggest problem relating to 'Hadrian's daughter' is then of course that Hadrianic Persecutions took place without Hadrian:

But Hadrian was not going to finish the war in person. Dio reports that when not only 'all Judaea' had been stirred up but even 'the whole world', Hadrian 'then, indeed, sent against them his best generals, of whom the foremost was Julius Severus.

— Birley

The story of the Ten Martyrs includes the name of the tenth martyr killed, Eleazar (ben Shammua). How the end of the Bar-Kochba-War is reported in some ancient sources, is just not reliable, with this one of the Martyrs —or one of his namesakes— meeting his end in quite a different way than in the poem, but also in relation to the emperor/Hadrian:

The decisive stage, not reported by Dio but by Eusebius, and well remembered, although in a very fanciful way, in the talmudic sources, came with a siege of the fortress of Bethar. It was 'a strong citadel', as Eusebius calls it, 6 miles (10 km) south-west of Jerusalem. The siege 'lasted a long time before the rebels were driven to desperation by famine and thirst and the instigator of their madness paid the penalty he deserved.'

A story in the rabbinic literature - which makes the siege last three-and-a-half years, perhaps by confusion with the duration of the whole war – has 'Bar Koziba' killing his uncle Rabbi Eleazar on suspicion of treachery.

'Forthwith the sins [of the people] caused Bethar to be captured. Bar Koziba was slain and his head taken to Hadrian … "Bring his body to me", he ordered.' It was found with a snake around the neck. 'Hadrian exclaimed: "If his God had not slain him, who could have overcome him?'"

No doubt it was Sextus Julius Severus, rather than Hadrian, to whom the dead leader was brought. It was probably in autumn 135, or perhaps early 136.

— Birley

Again we see quite some poetic licence at work, as this might refer to "Hadrian" as something not be read literally, if one insists on doing so, but "Hadrian's representative" or something like that. But that downgrades the reliability of the information to allegory or metaphor. If one wants to apply this to the info from Eleh Ezkerah and grant that text the same loose terminology as 'still historically correct info', the slope changes from just a slippery one to a chute into a rabbit hole.

The Eleh Ezkerah as seen in its modern form & English translation

The introductory notes for a modern English translation of the Midrash based on its most widely known version and the most complete one in Adolph Jellineks Beit ha-Midrash (1853) highlight:

The idea that ten sages were collectively executed by the Roman emperor is a legend conflated from several earlier martyrological sources. Talmudic and midrashic literature both narrate the separate deaths of sages who defied the edicts promulgated by the Roman emperor Hadrian against the Jews of Palestine following the brutal defeat of the Bar Kokhba revolt in 135 C.E.; these edicts included prohibitions against the study of the Torah and the observance of Jewish law.

Some of these accounts have been incorporated into our midrash, as have different versions of the martyrdom of sages believed to have died by Roman execution. But the lives of the ten sages cited in our midrash actually span a number of generations in the first two centuries of the common era. While a list of ten martyrs and the phrase “ten martyrs” (asarah harugei malkhut) are mentioned in rabbinic texts (Lamentations Rabbah 2:2 and Midrash Psalms 9:13), the earliest narrative about their deaths to anticipate our midrash is found in the early mystical-gnostic text, Heikhalot Rabbati, which recounts the martyrdom of Rabbi Ishmael and Rabbi Akiba, two eponymous heroes of early Jewish mysticism. The scholars of today agree that the legend of the ten martyrs began to take shape within these mystical circles, and the imprint of its origins is still visible in our midrash, whose lengthiest portion is devoted to the description of Rabbi Ishmael's ascent to heaven to learn whether the decree against the sages has been issued by God.

The legend of the ten martyrs is not, however, a mystical document, not even mystical hagiography. As martyrology, it shares more in common with pagan and early Christian martyrological literature than it does with other kinds of rabbinic writings of the period. Because of its legendary character, it is impossible to say exactly when the legend took on its present form.

— Stern, see below

The translation of the relevant passages read with ample evidently visible poetic licence:

This was the order in which the sages sat in pairs: Rabbi Ishmael and Rabbi Simeon Ben Gamaliel; Rabbi Akiba and Rabbi Hanina Ben Teradion; Rabbi Eleazer Ben Shammlja and Rabbi Yeshivav The Scribe; Rabbi Hanina Ben Hakhinai and Rabbi Yuda Ben Baba; Rabbi Hutzapit The Translator and Rabbi Yuda Ben Dema.

The Roman emperor now entered, followed by all the noblemen of Rome. He addressed the sages, “Who shall be executed first?” […]

So Rabbi Ishmael mourned and wept over the body of Rabbi Simeon ben Gamaliel. The emperor said to him, “What is this? You, an old man, weeping over your fellow sage! You should be weeping over yourself!”

Rabbi Ishmael replied, “I am weeping over myself! For my fellow sage was greater than I in knowledge of Torah and wisdom. And now he has joined the heavenly academy before me. This is why I am weeping.”

While Rabbi Ishmael was still giving voice to his grief, weeping as he lamented, the daughter of the emperor looked out her window and saw how handsome Rabbi Ishmael the high priest was. Compassion for him seized her, and she sent to her father asking him to grant her one request. He sent back to her, “My daughter, whatever you ask I shall do—so long as it does not concern Rabbi Ishmael and his colleagues.” She sent back, “I request you to allow him to live.” The emperor replied, “I have already taken an oath.” She responded, “Let me then request that you order the skin stripped off his face, that I may have it to use instead of a mirror to look upon myself.

— David Stern: "7. Midrash Eleh Ezkerah or The Legend of the Ten Martyrs", in: Victoria Clark, David Stern & Mark Jay Mirsky (eds): "Rabbinic Fantasies. Imaginative Narratives from Classical Hebrew Literature", Varda Books: Skokie, 2001. doi (A version of the 'Hekhalot Rabbati' to compare is here)