Do Christians and Muslims worship the same God?

So lets say you have a friend named Dave. and as we're talking, I mention my friend, who is also named Dave.

You look at me suspiciously: My Dave lives in Columbus.
I look back suspiciously: My Dave also lives in Columbus.

You smile: My Dave lives on High Street.
I grin: My Dave also lives on High Street.

You say: My Dave is a banker.
I say: My Dave is also a banker.

You say: My Dave drives a white car.
I say: My Dave also drives a white car.

You say: My Dave loves his son very much.
I say: My Dave doesn't have any children.

Is it possible we're talking about the same Dave? No. No it's not.

And say, "Praise to Allah , who has not taken a son and has had no partner in [His] dominion and has no [need of a] protector out of weakness; and glorify Him with [great] glorification."  (Quran 17:111)

Say: He is Allah, the One and Only; Allah, the Eternal, Absolute; He begetteth not, nor is He begotten; And there is none like unto Him. (Quran  112:1-4) 

It is not befitting to (the majesty of) Allah that He should beget a son. Glory be to Him! when He determines a matter, He only says to it, "Be", and it is. (Quran 19:35) 

“For God so loved the world, that he gave his only Son, that whoever believes in him should not perish but have eternal life.  (John 3:16)

Who has ascended to heaven and come down? Who has gathered the wind in his fists? Who has wrapped up the waters in a garment? Who has established all the ends of the earth? What is his name, and what is his son’s name? Surely you know! (Proverbs 30:4)

When we want to know if two people in question are the same person, like the two men named Dave in the opening example, we need to compare their attributes.  Are these two men exactly the same in every way?  We may find thousands of similarities between the two... they could both have beards, they could both have wives named Susan, they could both be graduates of the same university... but it is only necessary to find a single, irrefutable discrepancy to make the call.  In this case, I think, the teaching is very clear.