1 Peter, a servant and apostle of Jesus Christ, To those who have obtained a faith of equal standing with ours by the righteousness of our God and Savior Jesus Christ: 2 May grace and peace be multiplied to you in the knowledge of God and of Jesus our Lord. (2 Peter 1:1-2) [ESV]
1 Συμεὼν Πέτρος δοῦλος καὶ ἀπόστολος Ἰησοῦ Χριστοῦ τοῖς ἰσότιμον ἡμῖν λαχοῦσιν πίστιν ἐν δικαιοσύνῃ τοῦ θεοῦ ἡμῶν καὶ σωτῆρος Ἰησοῦ Χριστοῦ 2 χάρις ὑμῖν καὶ εἰρήνη πληθυνθείη ἐν ἐπιγνώσει τοῦ θεοῦ καὶ Ἰησοῦ τοῦ κυρίου ἡμῶν [mGNT]
This question is misleading as θεός is not used in either verse (it is found only in 2 Peter 2:4). The inflected term in verse 1 is θεοῦ: exactly as in verse 2. So from this narrow grammatical perspective, the answer is no; the same article prefaces the same word, these two terms are grammatically identical. It would appear to be a matter of theology whether τοῦ θεοῦ in verse 1 is cataphoric to θεοῦ in verse 2 or τοῦ θεοῦ in verse 2 is anaphoric to θεοῦ in verse 1.
However, if the meaning in verse 2 is different from that in verse 1, then semantically it might be anaphoric. The Lexicon supports considering this as a possibility:
❷ Some writings in our lit. use the word θ. with ref. to Christ (without necessarily equating Christ with the Father, and therefore in harmony w. the Shema of Israel Dt. 6:4, cp. MK 10:18...In 2Pt 1:1; 1J5:20 the interpretation is open to question (but cp. ISmyrna McCabe .0010, 100 ὁ θεὸς καὶ σωτὴρ Ἀντίοχος).
Since the first use of θεοῦ and is open to interpretation, different meanings between the two could justify treating one as primary and the other as anaphoric (or cataphoric).
Here is a comparison of the two phrases in question:
First, the pronoun ἡμῶν can be understood as making a distinction. That is, the God is anaphoric to the God of us. In this case Peter could be drawing a distinction between "God" in the general sense with "God of us" in a particular sense similar to Exodus 6:6-7, 2 Corinthians 6:16, or Hebrews 8:10. In context of the letter, the distinction is the difference between righteousness and knowledge. In others words, there is a knowledge of the God (in general) which is different from the righteousness of the God of us.
Despite the obtuse approach and creating a text in which τοῦ θεοῦ is open for interpretation, it is obvious the terms do not work to provide the interpretation suggested by the OP:
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 [the God from verse 2] and our Saviour Jesus Christ: 2 Grace and peace be multiplied unto you through the knowledge of God, [God is not Jesus here] and of Jesus our Lord.
In this analysis "our" must refer only to God: if the pronoun ἡμῶν is "relocated" to refer to "Savior" what remains is τοῦ θεοῦ, which is identical, not anaphoric. So the correct interpretation in this approach is the righteousness of our God and Savior Jesus Christ. This points to the second way τοῦ θεοῦ in verse 1 may be different from verse 2: τοῦ θεοῦ ἡμῶν and Savior is Jesus Christ exactly as the text states and which is affirmed by Sharp's Rule. [Ironically, the validity of Sharp's Rule to 2 Peter 1:1 is obvious from the OP's interpretation. The only justification for interpreting "our" as referring to Savior in the phrase τοῦ θεοῦ ἡμῶν καὶ σωτῆρος, is θεοῦ and σωτῆρος are referring to the same person.]
A grammatical analysis cannot be divorced from the writer's intent. Peter is not writing to give his readers the definition of τοῦ θεοῦ. He is writing to people who have already obtained the same faith by the righteousness of our God and Savior Jesus Christ. This means τοῦ θεοῦ ἡμῶν is anaphoric to what the reader already has. That is to say, Peter opens the letter by an anaphoric acknowledgement of τοῦ θεοῦ ἡμῶν the reader already possesses. The progression of "God" from verse 1 to verse 2, is the inverse of the readers prior experience:
Previous 1: knowledge of God...
Previous 2: equal faith by the righteousness of God...
2 Peter 1:1 equal faith by the righteousness of God...
2 Peter 1:2 knowledge of God...
Which simply reflects before they obtained the same faith of Peter they had a knowledge of God, but had not obtained righteousness. Until one makes a confession of faith in Jesus Christ, knowledge of God does not result in faith which saves. In other words, Jesus is our Savior only if one accepts Him:
9 because, if you confess with your mouth that Jesus is Lord and believe in your heart that God raised him from the dead, you will be saved. 10 For with the heart one believes and is justified, and with the mouth one confesses and is saved. 11 For the Scripture says, “Everyone who believes in him will not be put to shame.” 12 For there is no distinction between Jew and Greek; for the same Lord is Lord of all, bestowing his riches on all who call on him. 13 For “everyone who calls on the name of the Lord will be saved.” (Romans 10)
Those to whom Jesus Christ is Savior, call Him Lord, which is how Peter continues:
May grace and peace be multiplied to you in the knowledge of God and of Jesus our Lord. (1:2)
Peter is asking for grace and peace to be multiplied (a Pauline greeting) in the knowledge of two things God and Jesus our Lord. Very simply, in verse 1 Peter acknowledges the readers status as having been saved and in verse 2 is asking for them to have increase in the knowledge of God and the knowledge of Jesus our Lord. [Living as if Jesus is the Lord of you life is a key aspect of Peter's message.]
Peter makes two claims about Jesus Christ. He is our God and Savior. He is our Lord:
The essential "anaphoric" treatment within the letter is the chiastic device where ideas are repeated in reversed order: ABB'A'. Thus B' is anaphoric to B and A' to A.
For example, Paul uses a chiasm of grace and peace to bracket the letter to the Ephesians:
Grace to you (A - Ephesians 1:2a)
and peace. . . (B - Ephesians 1:2b)
Peace . . . (B' - Ephesians 6:23)
Grace. . . (A' - Ephesians 6:24)
Peter uses this same technique such that the opening statement is effectively repeated:
The closing, τοῦ κυρίου ἡμῶν καὶ σωτῆρος Ἰησοῦ Χριστοῦ refers back to the opening τοῦ θεοῦ ἡμῶν καὶ σωτῆρος Ἰησοῦ Χριστοῦ. Thus the structure in 3:18, "Our Lord and Savior, Jesus Christ" "brings back" the opening, "our God and Savior, Jesus Christ." The first statement (A) combines with the last statement (A') to repeat the opening proclamation, our God and Savior, Jesus Christ:
The God of us (A - 1:1a)
and Savior Jesus Christ (B - 1:1b)
the Lord of us (B' - 3:18a)
And Savior Jesus Christ (A' - 3:18b & 1:1b)
While it is true θεοῦ has different meanings in 2 Peter 1:1 and 1:2, the article in verse 2 is not anaphoric. Rather, the article in verse 2 serves to distinguish between God and Lord. It reinforces the emphasis from verse 1 where the article serves to show God and Savior are one. Taken together, verses 1-2 state since Jesus Christ is our God and Savior, He should be our Lord.
1. Fredrick William Danker, A Greek-English Lexicon of the New Testament and Other Early Christian Literature, The University Chicago Press, 2000, p. 450
2. James L. Resseguie, A Glossary of New Testament Narrative Criticism
with Illustrations, pp. 8-9