Be consistent in your reasoning. Transpose Muslims with Jews and try out your theology and you should see why your language is erroneous. It would be better worded as Islam does not believe God is triune; neither does Judaism, Mormonism, JWitnesses, Oneness Pentecostalism, etc. All these religions profess to believe in one God, the God of Abraham, Isaac, and Jacob.
The jews do not have a saving God either..
A bit more complex than the above. That's what the Muslims want people to believe. However, with Muslims Allah functions as the proper name of the deity. According to the Holy Scriptures, the one noun which functions as God's true and proper name is YHWH(Yahweh), not Allah.
YHWH made His Name known to Israel. YHWH is not the revealed Name in the Qur'an. Muhammad received his 'visions' or revelation from Satan or demons so we should never pony up to a Muslim and say we have the same God. Plus YHWH will not confuse and He set His Son's Name above all others.
The Qur'an does not place the Name of Jesus (Yeshua) Christ (Messiah) "which is above every name, that at the name of Jesus every knee should bow, of those in heaven, and of those on earth, and of those under the earth, 11 and that every tongue should confess that Jesus Christ is Lord, to the glory of God the Father.
The Qur'an is a false message and Muhammad a false prophet. Why should followers of Messiah Jesus try to even hint the Muslims worship the same God?