Mac mail does not work after 10.8.5 update
Grayed out Quit in Mail usually indicates that mail is busy doing something (like sync) and does not allow quitting.

Here are few steps you can do:

1- Run the Connection doctor

2- Enable the Activity window to see what is going on (get mail)

3- Delete that account and recreate it. (no worry, your email should be on the original server (ie gmail), but better check that before.

4- Wait for it to complete its duty. (like updating 10000 emails)