Short answer. Not in the way people mean. The Quran bans forced conversion, conquered peoples usually kept their own religion, and the world's largest Muslim populations were never converted by Arab armies at all.
Where the idea comes from
Within a century of the Prophet's death, Muslim rule stretched from Spain to Central Asia. That astonishing political expansion gets confused with religious conversion — as if every new subject was made to choose Islam or the sword. The history is very different.
The Quran forbids forcing faith
A foundational verse rules out compelled belief outright:
"There shall be no compulsion in [acceptance of] the religion. The right course has become clear from the wrong."
— Quran 2:256 (Saheeh International)1
"And had your Lord willed, those on earth would have believed — all of them entirely. Then would you compel the people in order that they become believers?"
— Quran 10:99 (Saheeh International)2
Consistent with this, conquered Jews and Christians generally lived as protected communities, keeping their own faith, law and worship under Muslim rule — which is why ancient Christian and Jewish communities survived for centuries across the Muslim world.354
Where Muslims actually live tells the story
If Islam had spread only by conquest, the most Muslim places on earth would be the lands the early Arab armies overran. They are not. Today the largest Muslim populations are in Indonesia, Pakistan, India and Bangladesh — and Indonesia, the single most populous Muslim nation, was never conquered by those armies. Islam reached the Indian Ocean world mainly through merchants, preachers and teachers, and conversion across the Muslim world was typically a gradual process over centuries.756
For the wider story of how Islam expanded, see A Short History of Islam and The Rightly-Guided Caliphs.
In short
Islamic empires grew partly through conquest, like every empire of the age — but faith was not imposed by the sword. The Quran forbids it, conquered peoples kept their religions, and most Muslims today descend from people who embraced Islam freely, through contact and conviction rather than coercion. See also Does Islam promote terrorism?.