Hello, 

I am working on the new update, it contains fixes for some problems with standard sound beep notification when new message comes to the messenger. First part already available on github.

About push notification: Push notifications should work on FF browser on Android, I am testing them also and it is working. There were problems with sound when notifications came to device, but sound works ok now also. 

Do you see any confirmation popups about getting notifications on messenger page when open FF browser in first time? Can you please try to remove existed notifications and try to open messenger page again and to allow notifications.

To clear the cache and remove push notifications you need to go to Settings -> Clear private data and select Cache, Site Settings, Offline website data, Site settings and press clear the data, then close the browser and open it again and login, it should show you new confirmation message.