When it comes to relations with Muslims, we must take care to prevent things from becoming any more complex than they admittedly already are, and we can do this by adopting as our starting point the position of the Catholic Church, which is that Catholics, even with their doctrinal differences, worship the same god as Protestants, Jews, Eastern Orthodox Christians, and Muslims.
Again, according to Nostra Aetate, it is not the god worshiped who differs, but the claims made about him by the worshipers:
“The Church regards with esteem also the Moslems. They adore the one God, living and subsisting in Himself; merciful and all-powerful, the Creator of heaven and earth, who has spoken to men; they take pains to submit wholeheartedly to even His inscrutable decrees, just as Abraham, with whom the faith of Islam takes pleasure in linking itself, submitted to God. Though they do not acknowledge Jesus as God, they revere Him as a prophet. They also honor Mary, His virgin Mother; at times they even call on her with devotion. In addition, they await the day of judgment when God will render their deserts to all those who have been raised up from the dead. Finally, they value the moral life and worship God especially through prayer, almsgiving and fasting.”
“[T]he plan of salvation also includes those who acknowledge the Creator. In the first place amongst these there are the Muslims, who, professing to hold the faith of Abraham, along with us adore the one and merciful God, who on the last day will judge mankind.”
In §841, the Catechism repeats the above statement.
Moreover, Pope St. John Paul II affirmed the same truth in a speech on May 9, 1985:
“As I have often said in other meetings with Muslims, your God and ours is one and the same, and we are brothers and sisters in the faith of Abraham.”
And, as if we needed further confirmation that this is a consistent teaching of the Church, we find it explained in Pope Benedict XVI’s Ecclesia in Medio Oriente that the connection between Christians, Jews, and Muslims is,
“grounded in the sacred Scriptures and clearly defined in the Dogmatic Constitution on the Church Lumen Gentium and in the Declaration on the Church’s Relation to Non-Christian Religions Nostra Aetate. Jews, Christians and Muslims alike believe in one God, the Creator of all men and women.”
That both Catholics and Muslims worship the god of Abraham is, therefore, a fact of Catholic teaching, and cannot be denied with any degree of seriousness.