Do Muslims and Christians worship the same God?

No. Muslims and Christians define God so differently that they are not describing the same God＊What this does and doesn't meanIs sayingThat the God Islam describes — one who has no Son — is not the God Christians worship, who is eternally Father, Son, and Holy Spirit; and that rejecting the Son does not leave you holding the Father: "No one who denies the Son has the Father" (1 John 2:23).Not sayingNot that Muslims are insincere, that Christians worship three gods, or that the two faiths have nothing in common — both are monotheists who look back to Abraham. Not a denial that a Muslim may be sincerely reaching for God: Christians have long distinguished who a person is reaching for from whether the God they describe is the true one. The claim is about who God is — not about the sincerity of the seeker, and not a verdict on any individual heart, which is God's to render. - and the difference is not a detail at the edges. It is who God is. Christians do not worship a solitary monad who happens to have a Son: the one God is eternally Father, Son, and Holy Spirit, and the Son is not an addition to God but essential to who he has always been. Subtract him and you are not describing a simpler version of the same God - you are describing someone else. A Muslim may sincerely seek the one Creator - but sincerity in the seeking is not the same as rightly describing the God who is really there, and reaching is not arriving. Some say the two faiths point at the same God and simply describe him differently; but what divides them is not a description laid over an agreed God - it is a denial of who that God eternally is.

Islam does not merely omit the Son; it denies him. In Pickthall's rendering the Qur'an tells Christians to 'say not "Three"' and puts a son far from God's transcendent majesty (4:171); and it insists that he 'begetteth not nor was begotten' (112:3). Scripture treats that denial as decisive rather than technical: 'No one who denies the Son has the Father' (1 John 2:23), and 'whoever does not honour the Son does not honour the Father who sent him' (John 5:23). John is not writing about those who never heard of the Son, but about those who met the claim and refused it - which is why this is no verdict on Abraham or Old Testament Israel, who lived before the Son was revealed. A God with no Son is not the God who so loved the world that he gave his only Son.

Two things are worth saying plainly. First, both faiths are monotheist, both trace their worship to one Creator, and both look back to Abraham - but a shared vocabulary and a shared ancestor are not a shared God. Second, at the popular level the objection usually misfires against a doctrine Christians do not hold: Christians do not worship three gods, and the church has ruled that out from its earliest creeds, which confess one God; no Christian creed has ever placed Mary in the Godhead; and the Trinity is not a partner bolted onto God from outside, which would indeed be shirk. It is what the one God eternally is within himself. The serious version of the objection is a weightier one - that the One cannot be internally distinct - and the linked study takes it on directly. None of this warrants coldness towards Muslim neighbours - it is precisely why Christians long for them to know the Father through the Son.