Short answer. No. Allah is just the Arabic word for God — the same God of Abraham worshipped by Jews and Christians. Arabic-speaking Christians and Jews use the word Allah too.

Where the idea comes from

To English ears the name Allah sounds like a foreign, separate deity, so it is easy to assume Muslims must worship someone other than the God of the Bible. A small number of writers have gone further and claimed that Allah was once a pagan "moon god." Both ideas rest on a misunderstanding of a single Arabic word.

"Allah" means "the God"

Allah is a contraction of the Arabic al-ilah — literally "the God." It is not a personal name like Zeus or Apollo; it is the ordinary word for the one Creator. Arabic-speaking Christians and Jews open their Bibles to the word Allah, because that is simply how you say "God" in Arabic.678 The Quran identifies this God as the very same one worshipped by the earlier prophets:

"And do not argue with the People of the Scripture except in a way that is best... 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 (Saheeh International)2

The heart of Islam is the oneness of God

Far from worshipping any created thing, Islam's first and greatest teaching is Tawhid — that God is one, unique, and unlike anything in creation:

"Say, He is Allah, [who is] One, Allah, the Eternal Refuge. He neither begets nor is born, nor is there to Him any equivalent."

— Quran 112:1-4 (Saheeh International)3

"There is nothing like unto Him, and He is the Hearing, the Seeing."

— Quran 42:11 (Saheeh International)4

This is exactly why the "moon god" idea collapses: the Quran explicitly forbids worshipping the sun, the moon, or anything else God has made.

"And of His signs are the night and day and the sun and moon. Do not prostrate to the sun or to the moon, but prostrate to Allah, who created them, if it should be Him that you worship."

— Quran 41:37 (Saheeh International)5

And the same call to the one God runs through the whole Quran:

"And your god is one God. There is no deity [worthy of worship] except Him, the Entirely Merciful, the Especially Merciful."

— Quran 2:163 (Saheeh International)1

For more on this God and how Muslims describe Him, see What is Islam? and The Names of God; on the shared lineage of faith, see Abraham in Islam.

In short

Muslims do not worship a different God. Allah is the Arabic word for the one God of Abraham — the same God of Judaism and Christianity — and the absolute core of Islam is that this God is one, without partner, and unlike anything He created.