Does the Issue Tracking support in NetBeans also work with netbeans.org Bugzilla?
Yes. Just open Services window (menu Window | Services) and add a new Issue Tracker. Specify https://netbeans.org/bugzilla as the URL.
See also Working with Issue Tracking in NetBeans IDE for more information.
Note: If you want, you can instead setup a new Team Server instance for netbeans.org and use not only the issue tracking but also other features. To do that, add a new Kenai instance (in the Services window, URL https://netbeans.org), make it the default, and login via menu Team | Kenai | Login. (This is available since NetBeans 6.8.)