Do Muslims and Christians believe in the same God?

In a basic sense, yes: both worship the God of Abraham. The Quran tells Muslims to say to Christians and Jews, "Our God and your God is one." Arab Christians even call God "Allah." The big difference is how God is described: Islam teaches strict oneness, while most Christians believe in the Trinity.

What the Quran says

وَقُولُوٓا۟ ءَامَنَّا بِٱلَّذِىٓ أُنزِلَ إِلَيْنَا وَأُنزِلَ إِلَيْكُمْ وَإِلَـٰهُنَا وَإِلَـٰهُكُمْ وَٰحِدٌ وَنَحْنُ لَهُۥ مُسْلِمُونَ "And say, 'We believe in that which has been revealed to us and revealed to you. And our God and your God is one; and we are Muslims [in submission] to Him.'" Quran 29:46

Muslims see Islam as continuing the message of Abraham, Moses and Jesus, all calling to the same Creator.

"Allah" is just Arabic for God

"Allah" isn't a special Muslim deity. It's the Arabic word for God, used in Arabic Bibles and by millions of Christians in the Middle East today. It's related to the Hebrew "Elohim" and the Aramaic "Alaha," which Jesus would have used.

Where Muslims and Christians differ

The Trinity: most Christians believe in one God in three persons. Islam teaches God is absolutely one, with no partners or son.
Jesus: divine Son for Christians; a great prophet and the Messiah for Muslims. See Do I have to believe Jesus isn't God to become Muslim?
Salvation: through Christ's sacrifice for many Christians; through faith in one God and good deeds, by His mercy, in Islam.

Scholars' nuance

Muslim scholars generally say both faiths intend the same Creator, but describe Him differently, and Islam holds that some Christian beliefs about God are mistaken. So "same God, different understanding" is a fair summary. Some Christian theologians say something similar from their side.

Why it matters for you

If you're a Christian exploring Islam, you're not abandoning the God you prayed to. Many converts say they feel they've found a clearer, simpler relationship with the same God. It also gives a strong basis for respectful friendships between Muslims and Christians. See Can I say "Merry Christmas" to my coworkers?

Key points

Both faiths worship the God of Abraham.
"Allah" is Arabic for God, used by Arab Christians too.
The Trinity is the main difference.
Same Creator, different understanding.

Tip: If you're from a Christian family, start conversations with what you share: one God, Abraham, Moses, Jesus and Mary. It makes talking about your journey much easier.