As of 12.04

Gtk3 is the default on 12.04.

For 11.x

Simple answer: No, it does not (Quickly 11.10).

Extended answer:
Since all the code produced by quickly can be edited, you can change it to support gtk3, but it does not come with support.

There is however a branch of quickly in development that (only) supports gtk3. You can see the difference in the templates: some have from gi import ... (which is needed for gtk3) and the older templates have import gtk (which does not work with gtk3).

You should be able to uninstall your current quickly version and get the gtk3-enabled version with:

bzr branch lp:~quickly-committers/quickly/gtk3-projects

I have however not tried it and it is of course not the current stable version of quickly (which has lots of implications...)