I get this question fairly often and I see it on the internet ALL THE TIME!
I was going to answer this myself, but then I happened upon an older post by Paul Anatar on the Kitchen and Residential Design blog and he answered the question far better than I ever could.
Plus, I think it's one thing for you to hear someone like me tell you not to install your own countertops, I'm making money off of having my company install them.
When someone not associated with a fabrication shop answers the same way, I think people tend to believe it a little more.
So, click here to read Paul's response of why you should not install your own quartz (or natural stone) countertops.