Who do Trinitarians believe is the Apostle Peter's God?

tl ; dr is Trinitarians can say 'God' is the Father, the Son, the Holy Spirit, or all 3, depending on the context. This is because God is whoever shares the divine substance, and all 3 persons share this substance. You can say each one 'is' God, or all 3 'are' God.

In the contexts above, they will say the Father is being referred to in specific, who is a divine person who 'is' God.

There's quite a bit more to the answer than just what has been proffered thus far by other users.

In Acts 1:1-7, we are told of an encounter with the post-resurrected Christ. In 1:1-2, the narrator identifies Christ as the one who had “chosen” (or appointed) the apostles to their position. In 1:6-7, that same Christ is referred to as “Lord.” And in 1:21, the subject is (again) referred to as “Lord Jesus.” Given that Christ has, by this time in the narrative, been identified as the one by whom the apostles received their position (1:1-2), and has twice thus been referred to as “Lord”; when we then come to the prayer lead by the Apostle Peter (1:24-25), there is no question to whom is being prayed to:

And they prayed and said, “You, Lord (cf. 1:6-7, 1:21), who know the hearts of all, show clearly which one of these two you have chosen (cf. 1:1-2, 9:15) to occupy this ministry and apostleship from which Judas turned aside to go to his own place”

In more than just one occasion (exclusive to the Book of Acts) is Christ the recipient of prayer: Acts 7:59-60, 9:10-21, 22:16.

Just as we heard in the earlier account (1:1-2, 1:24-25), Christ is said to have (Acts 9:15) chosen Paul as “my chosen instrument.”

In 9:11, the term “praying” (προσεύχεται) is left unqualified. The text doesn’t identify who the object of Paul’s prayer was, per se. Though, we can infer from the text who the recipient of that prayer was (notice the object v. recipient contrast). Paul may have directed his prayers to “the Father,” though it is also possible that he may have even been praying to “the Christ,” which he had encountered just moments before (9:1-9). Though, I do not find it all too likely that Paul would have addressed God as “the Father” in his prayer. It is likely that Paul may have neither invoked “the Father” or “the Christ,” but may have (in his moment of distress) invoked “the God of Israel” as the object of his prayer, or even perhaps invoked the Divine Name (or some substitutionary form), not anticipating that it was the risen Lord that was “hearing” the prayer. What is rather telling, is that in the very moment of Ananias’ vision, Christ says to Ananias (of Paul), “behold, he is praying (προσεύχεται),” using the Present Middle Indicative. For Christ to have known that Paul was (in that very moment) praying, and in an instance He (Christ) had also appeared to Ananias in a vision, strongly suggests that the risen Christ could hear Paul’s cry and plea for help in his moment of distress, whether that was Paul’s intended object or not.

And in 9:13-14, Ananias expresses great fear of Paul, exclaiming, “this man, how much harm he has done to your saints in Jerusalem, and here he has authority from the chief priests to tie up all who call upon your name!”

The expression, “all who call upon your name” is an allusion back to Acts 2:21 (“all who call upon the name of the Lord”), which is rooted in Joel 2:32 (cf. Acts 22:16). The OT is saturated with this sort of language, particularly in cultic/religious contexts, when prayer is offered up to none other than the God of Israel (cf. Gen. 4:26, Gen. 12:8, Gen. 13:4, Gen. 16:13, Gen. 21:33, Gen. 26:25, Ex. 34:5, 1Ki 18:24, 2Ki 5:11, Zep 3:9, Ps. 116:13, Ps. 116:17, etc). Given these intertextual connections, it seems out of place to suggest (should anyone do so) that Peter could not (or did not) invoke the Lord Jesus Christ as the direct object and recipient of his prayer in Acts 1:24-25.

Another point worth mentioning is that 1 Peter refers to Christ as “Lord,” following OT verbal patterns that were otherwise reserved when speaking of the God of Israel. For example, in 1 Peter 2:3, there is a clear allusion to Ps. 34:8 (“if indeed you have tasted that the Lord is good”). It is not coincidental that the author then goes onto allude to Ps. 34 later in his epistle (1 Peter 3:10-12). That Jesus is the referent here in 1 Peter 2:3 is brought out by what is said in the the very next verse, “to whom you are drawing near, a living stone rejected by men but chosen and precious in the sight of God” (1 Peter 2:4, Lexham).

In vv. 6-8, our author then goes on to cite a litany of intertextually related “stone” texts (Lexham):

For this is contained in Scripture:

“BEHOLD, I LAY IN ZION A CHOICE STONE, A PRECIOUS CORNER stone, AND HE WHO BELIEVES IN HIM WILL NOT BE DISAPPOINTED.”

This precious value, then, is for you who believe; but for those who disbelieve,

“THE STONE WHICH THE BUILDERS REJECTED, THIS BECAME THE VERY CORNER stone,”

and,

“A STONE OF STUMBLING AND A ROCK OF OFFENSE”

The first text (1 Peter 2:6) cited is a reference to Is. 28:16, “Behold, I lay in Zion a choice stone, a precious corner stone, and he who believe in Him will not be disappointed.” The second text (1 Peter 2:7) cited is a reference to Ps. 118:22, “The stone which the builders rejected, this became the very corner stone.” The third text (1 Peter 2:8) cited is a reference to Is. 8:14, “a stone of stumbling and a rock of offense.”

What may go unnoticed is that this final text which Peter cites (Is. 8:12-14 LXX), is with reference to “the Lord of hosts” as the “stone of stumbling and a rock of offense,”

μήποτε εἴπητε σκληρόν πᾶν γάρ ὃ ἐὰν εἴπῃ ὁ λαὸς οὗτος σκληρόν ἐστιν τὸν δὲ φόβον αὐτοῦ οὐ μὴ φοβηθῆτε οὐδὲ μὴ ταραχθῆτε κύριον αὐτὸν ἁγιάσατε καὶ αὐτὸς ἔσται σου φόβος καὶ ἐὰν ἐπ᾽ αὐτῷ πεποιθὼς ᾖς ἔσται σοι εἰς ἁγίασμα καὶ οὐχ ὡς λίθου προσκόμματι συναντήσεσθε αὐτῷ οὐδὲ ὡς πέτρας πτώματι ὁ δὲ οἶκος Ιακωβ ἐν παγίδι καὶ ἐν κοιλάσματι ἐγκαθήμενοι ἐν Ιερουσαλημ

Never say, ‘hard,’ for whatever this people says is hard, but do not fear what it fears, or be in dread; honor the Lord himself as holy. And if you trust in Him, He will become your sanctuary, and you will not encounter Him as a stumbling caused by a stone nor as a fall caused by a rock, but the house(s) of Jacob is in a trap, and those who sit in Jerusalem are in a pit.

Not only does Isaiah 8 identify the Lord of the OT as the “rock of offense,” but Peter even describes “Christ the Lord” following the very same verbal pattern that Isaiah 8 uses of that Lord when he writes,

1Peter 3:14-15
τὸν δὲ φόβον αὐτῶν μὴ φοβηθῆτε μηδὲ ταραχθῆτε, κύριον δὲ τὸν Χριστὸν ἁγιάσατε
(But do not fear what they fear, or be in dread, but honor Christ the Lord as holy)

Isaiah 8:12-13 LXX
τὸν δὲ φόβον αὐτοῦ οὐ μὴ φοβηθῆτε οὐδὲ μὴ ταραχθῆτε, κύριον αὐτὸν ἁγιάσατε
(But do not fear what it fears, or be in dread; honor the Lord himself as holy)

These are just a couple additional points of thought that I wanted to add in addition to what has already been stated by others regarding 2 Peter 1:1, 3:18. There is just one other thing that I would like to point out real briefly (which is often overlooked) regarding 2 Peter 1:1.

The KJV’s rendering of 2 Peter 1:1 is based on Thomas Beza’s Greek text of 1589 and 1598 that features an additional “our” (ημων) that is not found in the underlying Greek texts used in more modern translations; thus, the KJV reads, “our God and our Savior Jesus Christ” (του θεου ημων και σωτηρος ημων ιησου χριστου). This may also be the reason why other translations (especially those based on Beza’s Greek apparatus) render 2 Peter 1:1 in such way. As for the ASV, though not based off Beza’s Greek apparatus, it was heavily influenced by the KJV. It is possible that the ASV translation committee allowed—perhaps, even mistakenly—the KJV to influence their way of translating 2 Peter 1:1. Had they not referenced Beza’s Greek apparatus beforehand, it would be easy for someone to assume that the Greek text underlying the KJV reads the same as the Greek apparatus’ (Westcott-Hort and Tregelles Greek texts) underlying the ASV. And so they simply flip to 2 Peter 1:1 in the KJV without actually checking Beza’s Greek text. This is definitely not an unimaginable thing to do; we see it all the time. Hence, in James White’s publication, “The King James Only Controversy,” he chides the KJV for incorrectly translating 2 Peter 1:1, but he did not refer to Beza’s Greek apparatus which underlies it. Had he done so, I am certain he would have taken another approach. Unitarians are guilty of this too, because they will often times cite the KJV and other Bibles from that period that are based off a completely different Greek text, in an attempt to prove the plausibility of such a translation.

The reading found in Beza’s text has no textual support, and it’s internal evidence stands just as sharply against it. In every instance (aside from 2 Peter 1:1), Beza’s text places ημων in apposition to an articular phrase, but in this instance (2 Peter 1:1) it is Beza’s lone exception.

With the one answer given so far (as at November 2nd), you commented, "The word PERSON is not in the question, THEREFORE it MUST be defined in the answer." This was because the answer given dealt a lot with the Father being a person, the Son being a person, and the Holy Spirit being a person. That is the Trinitarian stance, one definition being that three persons subsist in the one being of God [or essence, or substance].

Most people who take Trinitarians to task fail to grasp the significant difference between 'person' and 'being', so that is why I am responding to your comment. Yes, the word 'person' needs to be defined (with respect to the one God the Bible speaks of.) The apostle Peter spoke of this one God, for he never spoke of 'gods' (plural) in such contexts.

The real issue here is the problem we have using English words that have been translated, first from old Latin, which, in turn, were translated from the original koine Greek. This is where misunderstanding arises. Here is how this is explained in the source below:

"Our English word 'person' comes from Latin and it has lost something in its translation from Greek. The original Greek word 'ousia' meant essence or substance. The usual Latin for that is 'substantia', meaning essence or substance.

The Greek word 'hypostasis' meant person, or a second meaning was substance. The usual Latin is 'persona', meaning person, or a second meaning of actor, or role.

The Greek word 'prosopon' meant face or mask, with a second meaning of person. The usual Latin for that is 'persona', and that is where we get our understanding of 'person' from.

But in the original Greek of the Bible New Testament, we see that 'essence' or 'substance' was the meaning, and it is that language that the Trinity doctrine is all about. [Most English speaking people today only think of it simply as an individual person.]

The Greeks described the Trinity as 'mia ousia en trisin hypostasesi' = one substance (essence) in three subsistenes [persons]." Unfortunately, that could be misunderstood as saying, "one essence in three substances", which would be 3 gods. When the Latins then said, 'una substantia in tribus personis' = one substance in three persons", they could be misunderstood as saying one 'hypostasis' (person) in three roles." [Bear in mind the link between a role and a face-mask, as with actors.]

We are 16 centuries removed from this, when Sabellianism opposed Trinitarianism. Yet the modern-day opposition and the teaching back then remains the same as today - "Three - Father, Son and Holy Spirit - are God, yet God is not three, but One." That is why I said that English translation from Latin loses something in the translation. And Latin translation from Greek lost something as well. There's nothing like sticking to the original koine Greek that the NT was written in, but few of us are linguists, and it was never going to be easy, getting a verbal handle on the awesomeness of Deity.

Source - Heresies and Orthodoxy in the History of the Church, Harold O.J. Brown pp 63, 128-130 (Hendrickson 1998)

Finally, to answer your actual question (now that I've exposed the tangle of misunderstandings we English-speaking people get into), for as long as there is not clarity with the meaning of translated words, some will insist Peter could only mean a simple God comprised of one person (or Spirit). Whereas others will insist Peter knew the one God to be a complex being, revealed in the incarnation, and through the teaching of the Holy Spirit.

Nobody will get anywhere asking, "What did Peter believe as to who his God was?" for what Peter said will be interpreted in light of one's theology. We all like to claim it is the other way around, with us - we discover what the Bible states and our theology is based on that. All I can say is that I stoutly and passionately believed in a simple God and that Jesus was created by this God, and the Holy Spirit was just energy, power, used by God. Then in later life, the Holy Spirit revealed to me just who this Son of God really is; that the Father and the Son share the one, divine nature, with absolute unity of the Spirit in that nature. I expect you to totally disagree, but that's not a problem here, on this site, as long as nobody starts trying to argue in comments that an answer is wrong.

Peter's belief about who God is VS Him addressing one person that is God

Obviously with the verses you have cited, Peter is addressing the Father, one person that is truly God, does this mean that Peter believed that God is only one person that is the Father, and not the Son and the Holy Spirit? No.

You have not lied to men but to God

Acts 5:3-4 (NASB95)

But Peter said, “Ananias, why has Satan filled your heart to lie to the Holy Spirit and to keep back some of the price of the land? “While it remained unsold, did it not remain your own? And after it was sold, was it not under your control? Why is it that you have conceived this deed in your heart? You have not lied to men but to God.”

Did Peter believe that the Holy Spirit is God? Yes, he showed that he understood the nature of the Holy Spirit to be God, it's clear that the "God" that is in verse 4 is applied to the Holy Spirit, notice the phrase "You have not lied to men but to God" connecting it with what he stated in verse 3 "to lie to the Holy Spirit" which indicates that he could not have addressed two persons in verses 4 and 3.

Our God and Savior Jesus Christ

2 Peter 1:1 (NASB95)

Simon Peter, a bond-servant and apostle of Jesus Christ, To those who have received a faith of the same kind as ours, by the righteousness of our God and Savior, Jesus Christ:

If you are using the KJV, it's not rendered clearly.

2 Peter 1:1 (KJV)

SIMON Peter, a servant and an apostle of Jesus Christ, to them that have obtained like precious faith with us through the righteousness of God and our Saviour Jesus Christ:

So did Peter call Jesus God? Yes he did, let's look at the context and the writing format of 2 Peter 1.

2 Peter 1:11 (NASB95)

for in this way the entrance into the eternal kingdom of our Lord and Savior Jesus Christ will be abundantly supplied to you.

Let's look at the Greek of the phrases in 2 Peter 1:1 and 2 Peter 1:11.

1:1

τοῦ Θεοῦ ἡμῶν καὶ Σωτῆρος Ἰησοῦ Χριστοῦ

1:11

τοῦ Κυρίου ἡμῶν καὶ Σωτῆρος Ἰησοῦ Χριστοῦ

You can see that Peter used the EXACT same format in 1:11 as 1:1 but used Κυρίου ἡμῶν (our Lord) instead of Θεοῦ ἡμῶν (our God), therefore, the rendering of "our God and Savior, Jesus Christ" is valid. This is called the Granville Sharp rule, where you have two nouns that are not general names such as John, Bob etc. and are connected by καὶ, and the first noun is supported by the definite article such as τοῦ in this case and the second noun is lacking one, which leads to the two nouns being applied to the one person that is addressed by name in the phrase.

So did Peter believe that God is one person?

No, as shown that Peter believed that the Father is God, the Son is God and the Holy Spirit is God, yet he was a monotheist. Citing verses where Peter is addressing one Person that is God such as the Father is a weak argument, I'm a Trinitarian and I pray to Father who is one person that is God and end my prayer in the name of the Lord Jesus.