J1 could have originated in Europe. At least, it could have existed in South East Europe, Ukraine and North Caucasus during the Epigravettian (Upper Paleolithic).
There are many theories concerning the Etruscans. I personally believe the original Tyrrhenians originated in Central Europe, but expanded in a region (Tuscany) where a language possibly related to the language of Mycenaeans was spoken. That is based on ancient sources like Dionysius of Halikarnassus. It isn't safe to trust ancient sources, but some people chose to believe what is written in the Bible for example. I think I will be ok.
Many would try to associate the Etruscans with ancient Anatolia. Therefore they would have had haplogroups found in Neolithic and/or Bronze Age Anatolia. But either way, most J1 subclades wouldn't have anything to do with them. Certainly, at least one subclade expanded with Semetic peoples. Sumerians could have had some J1. I am not very familiar with the phylogeny of any haplogroups though.