but why would anything relate to firefox, playing a video has nothing to do with it from your computer? if you have quicktime installed, if .mov files are associated with quicktime, double clicking them will automatically open quicktime on your computer and start playing
if .mov files are not associated with quicktime, should use whatever program on your computer is associated with it, nothing to do with the internet, ff, etc.
if not associated with quicktime, should be able to right click, 'open with' and select quicktime and it should play