Christians worship God in Trinity -- Father, Son and Holy Spirit. While Islam includes Jesus in its religious system and "honors" him as a great prophet (of Islam), it denies all the essential christological doctrines, and then has the gall to say that the Church has it wrong. Everything I affirm when I say the Creed during Divine Liturgy, Islam denies, except that Jesus is coming again. And, if I understand it correctly, Islam claims that the Comforter promised by Christ is not the Holy Spirit, but Mohammed himself.
So, no, looking at it this way, I would have to say that Muslims and Christians do not worship the same God.
Christians worship the God in Trinity. Jews and Muslims worship ONE God. So would you agree that Jews and Christians don't worship the same God?