It does not stand for quintessential.

How have I managed to get through life without questioning what the Q in Q-tips stands for?

And how have I managed to get through life without discovering that the Q stands for quality, which is completely stupid.

Quality tip?

Even so, it's better than the original name of the product, which was Baby Gays. 

I could find no explanation as to the origin of this name.

Oh, and just for the record, you are not supposed to clean your ears with Q-tips.  In fact, I have yet to meet a doctor who won't tell you that sticking anything in your ear is a bad idea. 

Even Q-tips own website does not list the cleaning of the inner ear as a recommended use.