No it doesn't because there's nothing in the term *God* that gives anyone any reason to think *Trinity*.
However, here by saying that *mother of God* makes starkly clear the reality of the Trinity equates to saying that Mary is the mother of the Trinity.
The Holy Spirit in the NT calls Mary *mother of Jesus*.
That is correct. But when we insist on Mary being the Mother of God when she gave birth to Jesus alone is a succinct statement of the nature of the Trinity, from which Christ cannot be separated.