If I understand your question, it's about whether you'll see the count increment over the bell icon over the top of the page, and that list of events when you click on it.
If that's the case, then I'm sorry, but presently there's no way to disable this.
If you're asking about desktop notifications via the browser, FF doesn't do or use this.
If you're asking about whether you receive e-mails upon certain events, there's only three events where you can get e-mails, and the first two are disabled by default:
1) When someone sends you a PM. However, the e-mail will only contain the screen name of the user who sent it - you would still have to login to the web site to read it.
2) When someone replies to a post or thread you wrote. However, it only appears to trigger on the first reply (if you wrote the thread), or on the first post immediately after a post you wrote in reply to a thread.
3) If you request a password reset. These e-mails can't be disabled, but in theory these shouldn't happen that often to begin with - only if you forget the password to your user account.
3 months
