John 8:6 would appear to show that Jesus could write:

John 8:6: This they said, tempting him, that they might have to accuse him. But Jesus stooped down, and with his finger wrote on the ground, as though he heard them not.

However, this passage, known as the Pericope Adulterae is regarded as inauthentic by the great majority of biblical scholars, being absent from most early manuscripts, as well as being found in different locations when it does occur. It provides no reliable biblical evidence that Jesus could write, and the presence of this verse could merely indicate a desire among later scholars to prove that Jesus could indeed read and write.

In The Birth of Christianity, pages 234-5, John Dominic Crossan cites evidence from rabbinic sources that support a conclusion that the literacy rate in the Jewish homeland was less than 3 per cent. He says this literacy rate, though low by modern standards, was not low at all taking into account the needs of a traditional society in the past.

Crossan points out Jesus was a peasant from a peasant village. Therefore, for Crossan, Jesus was illiterate until the opposite is proven. It is not proven but simply presumed by Luke, when he has Jesus read from the prophet Isaiah in the synagogue at Nazareth in 4:16-20 (Remembering that Luke, if he was the author of the gospel that now bears his name, appears never to have met Jesus). Jesus did not write, and in the opinion of Crossan, could not write.

First of all, we have to examine the use of the word "Rabbi". This Hebrew word means teacher, and in order to reach this level of knowledge and be a teacher, in addition to being trained in the OT scriptures, you had to be 30 years of age. This is why Jesus was 30 years old when he started his ministry. Then, we also have scripture that says the Religious leaders were amazed as his wisdom and that he was talking with these leaders when he was a boy.
Luke 21:37 says that He taught everyday at the temple. English Standard Version "And every day he was teaching in the temple, but at night he went out and lodged on the mount called Olivet."

This passage is significant because it shows that this was a regular thing for Christ, and secondly, Luke himself was a physician, and Luke has long been accepted by scholars around the world as a world-class historian.

Then, we have more evidence from scripture of his exceptional wisdom and learning in Luke 2:41-51.
46 Three days later, they found him in the Temple sitting among the teachers, listening to them, and posing questions to them. 47 All who heard him were amazed at his intelligence and his answers.

Furthermore, we have other texts of Christ reading from the scrolls in Isaiah.
**This is confirmation of not only the fact that he was schooled in the OT, but these scrolls did not have chapter and verse marks. These were added for convenience hundreds of years later. One had to be so well-trained in the scrolls, that he could find one section out of the entire book in a scroll. In Luke chapter 4, Christ was in the synagogue in Nazareth, his home town, and he was given the Scroll of Isaiah. "In Isaiah 61, there are passages that speak of an anointed one who preaches the good news to the poor, frees the people who are imprisoned, heals the blind and releases the oppressed. About 700 years after the time of Isaiah, Jesus relates these Bible passages to Himself". Out of all of Isaiah, he found the specific prophecy that was about the Messiah - about him, and he read it, and then told the religious leaders and the others at the synagogue, "Today, this scripture is fulfilled in your hearing."
He not only was well-educated, and could read and write, but he also could find one section out of a scroll which is very large and extremely complex [66 chapters] and he also quoted from other OT prophets by heart, from Jonah, and from the Torah.

Source: https://bible.knowing-jesus.com/Luke/2

http://www.aboutbibleprophecy.com/isaiah_61_1.htm

No

Excellent question, brother!

John 7:15. 15 And the Jews marvelled, saying, How knoweth this [Jesus] letters, having never learned [scriptures]?

At that time, very few people were literate. Fewer yet that were carpenters. For Jesus to have been literate and having "never learned" scripture is very unlikely.

But, it's totally fine. There is nothing about my G*d that necessitates literacy. And with so many good and decent ardent supporters that are illiterate even today, it seems like it may have been strategic for G*d to have endowed his only son with the kind of ignorance that the common man could relate to.

Moreover, even more contemporary religions like Islam and LDS have illiterate prophets leading them, and there was even less of a demand for literacy in the era of The Son.

The question asks for biblical evidence. This turns out to be mixed. There is evidence in the gospels that Jesus was illiterate, and there are also clear statements that he was literate. We probably have no way of knowing whether these were two disagreeing threads of preserved contemporary memories, or whether the claims of literacy were a later overlay.

Some scriptural evidence against literacy comes from the passions of Mark and Matthew. Jesus's final words according to Mark 15:34 and Matthew 27:46 are a quotation from Psalms, in Aramaic, which suggests that these two evangelists thought that his knowledge of the scriptures was in Aramaic. This would make sense if Jesus was illiterate, because an illiterate Palestinian Jew at this time would have heard the scriptures orally in Aramaic. These translations, called targums, were not written down until after the destruction of the Second Temple. During Jesus's lifetime, only the Hebrew version was written down.

In favor of literacy, passages such as the following clearly state that Jesus could read the Tanakh (Hebrew), and he had excellent reading comprehension (which my neighbor the kindergarten teacher will assure you is not guaranteed just because you can read). He would have to have been multilingual in Aramaic, Hebrew, and possibly Greek.

Mark 2:25

He said to [the Pharisees], "Did you never ἀνέγνωτε what David did when he had need and was hungry -- he, and those who were with him?"

The verb ἀνέγνωτε literally means "re-recognize," and has a variety of possible meanings, including to know something well or to read it out loud. This passage clearly implies that Jesus's knowledge is superior to that of the Pharisees, but it doesn't necessarily refer to reading.

Luke 4:16

He came to Nazareth, where he had been brought up. He entered, as was his custom, into the synagogue on the Sabbath day, and stood up to read. 17 The book of the prophet Isaiah was handed to him. He opened the book, and found the place where it was written,[...]

Here it's clear that he can read a scroll. The following passage can be read either way, as evidence for or against Jesus's literacy.

Mark 6:2 (near-identical text at Matt 13:54)

When the Sabbath had come, he began to teach in the synagogue, and many hearing him were astonished, saying, “Where did this man get these things?” and, “What is the wisdom that is given to this man, that such mighty works come about by his hands? Isn’t this the carpenter, the son of Mary and brother of James, Joses, Judah, and Simon? Aren’t his sisters here with us?” So they were offended at him.

It's pretty clear here, although not stated explicitly, that Jesus's education is unexpected given his class background. Teaching in a synagogue does not necessarily imply literacy. From the context, this passage probably describes events in Nazareth (not Capernaum). Any synagogue in Nazareth would have been small, probably not much more than a small, humble building, possibly with some scrolls. The people in attendance would have been illiterate peasants who spoke only Aramaic, and there were no written targums at this time, so Jesus would have been delivering his teachings orally, from memory, in Aramaic -- not reading aloud to an audience in Hebrew.

Although literacy was not widespread among working-class people like Jesus, there were rabbis, and they could read. If Jesus was literate, then we learn here and in other places in the gospels about his level of literacy. Taking these depictions at face value, it would be clear that Jesus is a master debater, so that his understanding of the scriptures is very deep -- not the kind of partial or garbled understanding that you would expect for someone who was barely literate or had poor reading comprehension.

But actually the passages in the gospels where Jesus quotes scripture, shows deep scribal literacy, or engages in technical legal debates often show signs of being later layers added by Christian writers. For example, Luke 2:46 has a three-day delay before Jesus's family finds him at the temple, and this is later Christian symbolism. In more clearly authentic passages, Jesus's style is to teach in parables, cite secular proverbs, and make up his own short, pithy sayings.

The Bible doesn't clearly say whether he could write.

John 8:6

But Jesus stooped down and wrote on the ground with his finger.

The verb translated here as "wrote" is κατέγραφεν, which could mean either that he wrote or that he drew.

So in summary, the biblical evidence is contradictory.

As far as extra-biblical arguments, people like John Dominic Crossan, Chris Keith, and Reza Aslan have written at great length on this topic and haven't come up with anything conclusive, and they don't agree except to agree that their conclusions are tentative. Jesus had no reason to write down his teachings, because (1) nearly all his followers would have been illiterate, (2) his community embodied his teachings, and (3) he taught that the world would end imminently, so that there would be no future generations who would need a written record.

Times change, and the exact measure that defines "literacy" changes with them. The concept of an "essay" would have been unknown in Jesus' day, but it is quite clear from the record of we have of his interaction with other learned men that he was treated as if he was educated. As we learn from the verses quoted in parap's answer he was clearly able to read the Tanakh scrolls. These two factors are enough for us to label him as being literate. Living and ministering in a primarily oral culture, it is not surprising that Jesus chose not to sit and write books but rather preached with parables to the public and by reasoning from the Scriptures with those educated in them.

Matthew 13:54 (ESV)
⁵⁴  and coming to his hometown he taught them in their synagogue, so that they were astonished, and said, “Where did this man get this wisdom and these mighty works?

The verse here says that Jesus astonish them learned men of his day with his wisdom and he taught them. This is not the work of an unlearned man. Teaching from, reasoning from and accurately quoting the OT Scriptures (dozens of times) are clear signs of solid education and literacy skills.

Jesus read Isaiah 61 in a synagogue:

Luke 4:16-22 (ESV)
¹⁶  And he came to Nazareth, where he had been brought up. And as was his custom, he went to the synagogue on the Sabbath day, and he stood up to read. ¹⁷  And the scroll of the prophet Isaiah was given to him. He unrolled the scroll and found the place where it was written, ... ²⁰  And he rolled up the scroll and gave it back to the attendant and sat down. And the eyes of all in the synagogue were fixed on him. ²¹  And he began to say to them, “Today this Scripture has been fulfilled in your hearing.” ²²  And all spoke well of him and marveled at the gracious words that were coming from his mouth. And they said, “Is not this Joseph's son?”

Jesus wrote on the ground:

John 8:6-8 (ESV)
⁶  This they said to test him, that they might have some charge to bring against him. Jesus bent down and wrote with his finger on the ground. ⁷  And as they continued to ask him, he stood up and said to them, “Let him who is without sin among you be the first to throw a stone at her.” ⁸  And once more he bent down and wrote on the ground.

In addition to the direct implications of these, we have less direct but still strong evidence in events such as his delivery of the Sermon on the Mount. (Matthew 5-7). Also, if you are a Christian, you believe he co-authored the entire Bible (Hebrews 1:1-2).