Those who say Allah refers to the same God as Christian theism sometimes use this argument: Muhammad had some knowledge of Jews and Christians. He may have gotten his information from members of the Jewish community as well as his Christian uncle (Waraqa Ibn Nawfal). So he intended to refer to their God.
But I think that argument misses the target. Non-Christians can certainly refer to the Christian God. No doubt non-Christian theists like Maimonides, Al-Ghazâlî, Dale Tuggy, and Seyyed Hossein Nasr (to name a few) can and do refer to the Christian God. (For that matter, atheists can refer to the Christian God.) They talk about the Christian God when they attempt to critique Christian theism.
That, however, is not to be confused with the question of whether their God refers to our God. Indeed, they typically contrast their God with the Christian God.