No, it’s not possible to change your nickname in Quake Live. The only thing you can change about your nickname is the following:
- Your clan tag (5 characters maximum, use the command /clan in the Quake Live console)
- Capital letters or lowercase letters
However, what you can do to get a new nickname is create a new account in Quake Live (not possible if your account is linked to Steam).
Id Software does not care too much about duplicate accounts. Quake live does not exclusively use the number of registered accounts as a reference for traffic and popularity. The userbase statistics would be very inaccurate which would not convince potential advertisers. Don’t spam too many new accounts either though, just as a precaution.
Read these Quake Live config customization tips for many more useful console commands.