I read that the Mesopotamian culture as far back as 2000 BCE was highly patriarchal and got to a point where women could not talk to men outside their family and had to wear veils. Their are very similar concepts today in some Islamic States. How much of these concepts can be attributed to Mesopotamia or is it all entirely based on the Quran?
Islam is an interesting religion. Whereas Christianity is often neatly complied in a single document, the Quran is by no means the single word of authority for the Ummah (Muslim community.)
There are also documents called 'Hadiths' which are little writings which could be written by anyone, pretty much, and Islamic scholars made up their own mind as to whether or not it was good enough to be considered 'canon'. The Quran itself was revolutionary in its treatment of women, and hugely advanced the rights of many women, setting out the idea that women should be able to work, and that inheritance should be received by women too. Some of the ultra-patriarchal aspects were added later.
It is entirely possible that some of the more patriarchal aspects of Islamic law were written by people who were influenced by the Mesopotamian tradition of a stronk patriarchy. So, to that extent, yes, some of the Islamic patriarchal traditions could have their roots in ancient Mesopotamia.
However, there are also competing reasons for this. One is that Muhammad (pbuh) would get his wives to wear a veil when they went out so their honour wouldn't be compromised, I guess a bit like the way Michael Jackson didn't let his kids show their faces in public, and some suggest it just sort of caught on.
Another idea is that it was already common before the rise of Islam in Arabia, many tribes would have been patriarchal, and its likely some forced women to wear veils. A member of these tribes could have contributed to Islamic canon with his little brand of patriarchy.
Finally, its likely that it's just a result of the times. The patriarchy was big in the pre modern old world, and full face veils were as prevalent in medieval Christendom as in the Muslim world, and so it might just be a result of over conservatism in the modern Islamic world that this tradition has stuck around, whereas it died out in Europe.
As the other poster says, attributing this all to the Qu'ran is a mistake and misunderstands Islamic tradition. I would suggest, however, that going back to Mesopotamia is a bit of a stretch. Mesopotamian society, as I believe you are using the term, is separated by both considerable time and space from where islam was founded in the Arabian peninsula.
As for why women would wear veils in Islam, Asma Barlas attributes it to a marker protecting women from sexual assault by showing that they were not slaves, and hence free to be taken advantage of.