Since Trello doesn't yet support desktop notifications for Firefox, I decided to build an add on that provides support for it.
Usage instructions: As of version 0.1.2 the addon will take you to the login page for trello upon being installed. Give permission to the app and the tab will close automatically after which you'll get the notifications as toasts. To log out, click the right icon and the notifications will stop till you log in again. To login again, just click the bell icon on the left side. This still isn't ideal, and a new version will bring the whole add on into a single icon with a web page for configurations.
Notes on Support: I haven't added full support for all the notification types that trello has. These will be released in what I hope will be a rapid set of updates to the add on.