The question of whether or not the God of the Christians and the God of the Muslims is the same is a controversial one. For many Christians, the answer is quite easy. Christians believe in the Trinity, Muslims reject the Trinity, therefore they are different Gods. But is it that simple? Much could be said about the nature of the Trinity and what Christians actually believe and what Muslims actually reject. However, Miroslav Volf helps bring some perspective by looking at Christian attitudes toward Jews.
“Do Christians likewise conclude that Jews do not have the same God they do? Overwhelmingly they do not. Neither Jewish rejection of the doctrine of the Trinity nor the accusation that Christians are idolaters has led Christians to assert that Jews believe in a different God! That would turn Jews into idolaters. … From the Christian side, the debate with Jews was about how to describe God properly (whether to do so in trinitarian terms or not) and how to worship God properly (whether God should be worshipped through Christ or not). The debate with Jews was never whether Jews and Christians worshipped the same God.”
This does not prove that Christians and Muslims worship the same God. But it does suggest that we need more than simplistic answers.