Which is the BEST Open source tool for QA Management, which can provide integration with Bugzilla and jira.
It's probably worth checking out Tarantula which is available as a free open source test management tool that can integrate with Jira and Bugzilla.
Testlink, Open Source Test Management system. It can integrate with a variety of bug management tools as well as automation tools.
Search for the Testlink tag in this site and you will see numerous discussions regarding it, if you require more information or you can browse their website for more information. You can access their demo site, if you would like to see it in action via the following link TestLink Demo site and use admin (username) and admin (password) to login.