Firefox has no email functions by itself. If you do have some email functions in Firefox, they were added by an extension and you should seek support for that extension through the developer of that extension.

How do I find out what this extension was? This functionality (as far as I can tell) was added by my simple action of choosing foxfire as my default browser and creating a gmail account. It existed on my HP pavilion desktop with Windows Vista without my doing anything to set it up.

Probably one of these extensions.
https://addons.mozilla.org/en-US/firefox/search/?q=Gmail+Notifier&cat=all&x=0&y=0

Thanks for the help. The only one which worked was Gmail Watcher which does not change the icon. But it provides similar functionality by creating a periodic message visible at the bottom left of the screen telling me how many unread emails are in my inbox. The ones I tried which did NOT work were: Gmail notifier by Kang; Gmail Manager by Long Gmail Checker by Nguyen; and Gmail Favicon Notifier by Pendelton.