You can now enable animated screennames in your account preferences > Edit > Screenname > Animated Image.
Once enabled it will use such animated images for user screennames. Just be warned, depending on your browser and/or hardware this can slow down your daily experience on this website.
It is disabled by default for newly created accounts.
You can find out more about how to style your screenname here. Basically you can copy over your in-game Quake nick and enjoy it here.