The Mesopotamians had their own writing system. The Mesopotamians are said to have invented the wheel. The reconstructed vocabulary of the Proto-Indo-Europeans indicates that they had wheels. There is no vocabulary that I know of that indicates writing (they could have a word that means "to record" or something like it), but the way that the wheel spread made me think the Proto-Indo-Europeans had a simpler, older version of cuneiform.
A number of people lived in Mesopotamia, including the Sumerians, Kassites, Akkadians, Amorites and other Semitic-speaking people, and Persians. Cuneiform writing was invented by the Sumerians, and widely adopted by many cultures including the Iranians and the Hittites (and their kin), a number of years later. But Indo-European presence in Mesopotamia (which is different from Anatolia) is a more recent and limited historical development. It is implausible that they had a writing system which left no trace, inspired the Sumerian development of their hieroglyphics (whence cuneiform), then re-adapted it millenia later when the Hittites picked up this writing system from Assyrians to the south-east.