What the Qur’an says
The Qur’an tells Muslims how to speak to Jews and Christians, and it begins from common ground:
وَإِلَٰهُنَا وَإِلَٰهُكُمْ وَاحِدٌ وَنَحْنُ لَهُ مُسْلِمُونَ ٤٦
والهنا والهكم واحد ونحن له مسلمون…and our God and your God is one; and we are Muslims [in submission] to Him.
The verse does not say the two communities believe the same things about God. It says the one they address is the same.The quotation is the end of a longer verse that begins with an instruction about how to argue with the People of the Book.
Where the disagreement lies
The Qur’an’s criticism of Christian doctrine is a criticism of description, not of address. Calling God three, or ascribing a son to Him, is what the Qur’an names shirk: associating partners with God.
So, the same God?
In intention, yes: there is no second Creator to worship. In description, Muslims hold that Christian doctrine says things of God that are not true of Him. Both statements are needed for an honest answer.