I guess this depends a bit on how you would define a civilization. I would say that the most likely answer -- the one you'll see most -- is that civilization developed in Mesopotamia with the Sumerians. Their dates are generally given as somewhere in the 3000 to 2000 BC. But this date is given as the start of when they became urbanized so you would only use that if you are saying that civilization starts with cities.
If you would say that civilization starts with agriculture, then you are looking at a much earlier date. Jared Diamond talks about dates around 8500 BC. The eNotes "Salem on History" summary mentions a date of about 6000 BC.
There is a lot of ambiguity in trying to define exactly when a civilization began and so you sort of have to pick a number based on what you think civilization consists of.