Yes and No....
Both Muslims and Christians do worship the same God... Muslims refer to him as Allah whilst majority of Christians call him GOD... but in the bible Jesus (may peace be upon him) has been known to call him "Ellah".
The part where Christians differ is that they also consider Jesus God. They go as far as calling him son of GOD... Muslims only revere Jesus as a messengers exactly like they revere of Muhammad, Moses,, Abraham, Noa etc. (May Peace be upon them all).
(sorry for any English mistakes, because it's only my second language)