Q has nothing to do with mIRC directly.
If you're getting the registration dialog at the start, then your copy of mIRC is not registered correctly. You can also check Menu, Help, About mIRC. It should say "Registered to" at the bottom left.
Q is a Services system, probably on QuakeNet, though I don't know if any other networks run a Service called Q. If you're having problems identifying with Q, your best bet is to ask help in the #Help channel.
