No, Matthew 1:22, 23 does not imply that Jesus is God.
A fairly common trinitarian “proof” that “Jesus is God” uses the meanings of personal names. It is very true that personal names were extremely important to God and to his people as recorded in the Bible. The meanings of their names were often carefully selected by their parents and were sometimes changed during their lifetimes because of changing circumstances.
But honestly, should Jesus really be considered to be God because he was symbolically “named” Immanuel (Is. 7:14; Mt. 1:23) which means “God is with us”? No more so than Gabriel was calling himself God when he visited Mary and declared: “The Lord is with thee” - Luke 1:28. Nor did Zacharias mean that John the Baptizer (his new son) was actually God when he was asked, “I wonder what this child [John] will turn out to be?”, and he answered, “Praise the Lord, the God of Israel, for he has come to visit his people and has redeemed them.” - Luke 1:66-68, LB.
Gabriel and Zacharias (Zechariah) meant exactly what Israelites have meant throughout thousands of years when saying “God is with us” and similar statements. They meant “God has favored us” or “God is helping us”. - Joshua 1:17; 1 Samuel 10:7; 2 Chron. 15:2-4, 9 (cf., Jer. 1:8; Haggai 1:13). But if we insist on trinitarian-type “proof,” then Gabriel must have meant that he (Gabriel) is God. And Zacharias (whose own name means ‘Jehovah is renowned’ - p. 678, TDOTB) must have meant that John the Baptizer is God! – Also see 1 Sam. 17:37; 2 Sam. 14:17; 1 Ki. 8:57; 1 Chron. 17:2; 22:18; 2 Chron. 1:1; 35:21; 36:23; Ezra 1:3; Is. 8:8, 10; Is. 41:10; Amos 5:14; Zech 8:23. (Also see “Immanuel” in the Insight books.)
This understanding is seen throughout the Bible. For example, “But if all prophesy, and an unbeliever or outsider enters, he is convicted by all, he is called to account by all, the secrets of his heart are disclosed; and so, falling on his face, he will worship God and declare that God is really among you.” - 1 Corinthians 14:24-25, RSV.
Or, in a Psalm many of us apply to ourselves or our friends:
4 Yea, though I walk through the valley of the shadow of death, I will fear no evil; for thou art with me - ASV.
Even the widely acclaimed trinitarian Bible dictionary, The New International Dictionary of New Testament Theology, 1986, Vol. 2, pp. 86, 87, admits:
“The name Emmanuel [or Immanuel] which occurs in Isa. 7:14 and 8:8 means lit. ‘God [is] with us’ .... In the context of the times of Isaiah and King Ahaz the name is given to a child as yet not conceived with the promise that the danger now threatening Israel from Syria and Samaria will pass ‘before the child knows how to refuse evil and choose the good.’ Thus, the child and its name is a sign of God’s gracious saving presence among his people .... [The name Emmanuel] could be a general statement that the birth and naming of the special child will indicate that the good hand of God is upon us.” - p. 86. And, “The point of the present passage [Matt. 1:23] is to see in the birth of Jesus a saving act of God, comparable with the birth of the first Emmanuel. Both births signify God’s presence with his people through a child.” - p. 87.
For more, see:
This identity of Jesus Christ as Immanuel did not mean he was the incarnation of God (Insight-1 pp. 1187-1189; Watchtower online Library)
Does Matthew 1:23 indicate that Jesus when on earth was God? (rs p. 209-p. 220; Watchtower Online Library)
Exposing the False Reasoning Behind Trinity Proof Texts (Examining the Trinity)