Greetings in that Divine and Most Precious Name of Our Lord and Savior Jesus Christ!
This is where Christology is very important and crucial to understand this.
The simplest answer is yes, we uneqouvically worship the same God, and we even use the same terms, Arab Christians and Muslims alike call to the Lord as "Allah"
Further, we fundamentally believe it is the same God. However, where Christianity is unique, different, and most accurate is in the Christological realities of the Incarnation.
Jesus Christ, born of the Virgin Mary, is the Incarnation of the Word, of the Divine Logos. Through the Incarnation and ONLY through the Incarnation, what was previously invisible, immaterial, intangible, and basically innaccessable, the Divine, was made available to human beings in the form of deified human flesh. Jesus Christ unveils the Father and the Divine, because He is such. Before the Incarnation, the gulf of sin and material physicality separated us from God, and the Law (of both Moses and Muhammed) are mechanisms to try to make amends for the gap, but can never of themselves bridge this gap. Only God Himself, taking on flesh and blood, assuming a human form and nature, bearing the cross and death and resurrection, gives us access to the Godhead. So we know in the fullness of faith that we worship God in the fullest manifestation, and that aside from the Union of the Incarnation, we have no true aspect to the Divine.
So yes, in concept Muslims worship the same God, but since they do not worship Him in the hypostatic flesh and blood in the Person of Jesus Christ, they have no direct access to the form, to the substance, to the hypostasis of the Godhead. So they worship that which they cannot actually touch or access or know in a real, concrete sense. Whereas we Christians worship God in the flesh, who we know, who we touch and receive, and so we in the most literal sense know God, and Christology suggests that we can ONLY know God through and of this way, and so Muslims and Jews, now after the Incarnation, have denied the only access to God we have been offered.
So in theory we worship it same God, but not in form or reality, because Muslims by definition deny the manifestation of God in our midst, however even amongst Christian heresies we have had similar problems with accepting this, and so many heretic groups are equally denying access to the Godhead in the truest sense. Only God can reveal Himself, and God does so through His form in the Incarnation, and so we must come to God and meet Him directly to know and worship Him in a personal way, in a relationship. Essentially, Muslims worship an impersonal aspect of God, and so can not actually know Him, where as we know and love Him exactly and more directly as a Person.