GMail uses a old draft version of the API, and they check for the vendor-specific window.webkitNotification rather than the spec window.Notification, so a Firefox add-on or a GMail update is necessary.
Re:Desktop notifications in Firefox nightlies
by daleharvey on Saturday March 23, 2013 @02:28PM
(#5429131)
Chrome has an old implementation but as far as I can tell also supports the new one
http://jsbin.com/notification/1/edit works in chrome and firefox
If apps are using the old spec they may need to switch before both browsers work
Is this API compatible with Chrome's implementation, or would Google have to add support in Gmail for Firefox?