What civilization began in the country that is now known as Iraq?

Mesopotamia.

Mesopotamia is a region, not a civilization. Numerous civilizations started in what is now known as Iraq and are considered the Mesopotamian Civilizations like: Sumer, Akkad, Chaldea, Assyria, and Babylonia. It is also notable that the Abbassid Caliphate was created and headquartered in Iraq although it was a successor state to the Umayyad Caliphate based in modern Syria.