Bzzzt. Wrong! Muslims reject the triune God of Christianity and they reject Christ as the Son of God. This automatically MEANS that they do not worship the same God as Christians. And logic requires that the muslim view of God and the Christian view of God CANNOT BOTH BE TRUE. Your statement is thus proven absurd.
I am the last one to defend Islam, but by your definition of what constitutes the "Christian" God, you would therefore postulate that the Jews don't worship the same God we do, simply because they don't accept Christ as the Son that of God.
YOUR statemnt is both absurd and religiously bigoted. Read some books about comparative theology. Allah, Yahweh and "God" are the same person whether or not you accept Christ as his son.