Firefox account isn't for backup option - sync is something that uses Firefox account to sync between devices. But if this was standalone and you didn't do a Firefox data backup following Firefox FAQ on how to do this anything that was on the Gateway before the install is now gone and there is no recovery or restore.

Were you using Sync on the old device ? If you did then you would merely have to connect the new device to this Sync account using the same email and password.

• https://support.mozilla.org/en-US/kb/how-do-i-set-sync-my-computer

you can also transfer files directly to the new device.

You can copy certain files with Firefox closed from the profile folder on an old device to the profile folder on a new device to transfer your personal data. Note that best is to avoid restoring a full profile folder and only restore important files that are safe to restore/transfer.

You can use the button on the "Help -> More Troubleshooting Information" (about:support) page to go to the current Firefox profile folder or use the about:profiles page (Root directory).

• Help -> More Troubleshooting Information -> Profile Folder/Directory:
  Windows: Open Folder; Linux: Open Directory; Mac: Show in Finder
• https://support.mozilla.org/en-US/kb/profiles-where-firefox-stores-user-data

• bookmarks and history: places.sqlite
• favicons: favicons.sqlite
• bookmark backups: compressed .jsonlz4 JSON backups in the bookmarkbackups folder
• cookies.sqlite for the Cookies
• formhistory.sqlite for saved autocomplete Form Data
• logins.json (encrypted logins) and key4.db (encryption key and primary password) for logins saved in the Password Manager
• cert9.db for certificates stored in the Certificate Manager
• persdict.dat for words added to the spell checker dictionary
• permissions.sqlite for Permissions and possibly content-prefs.sqlite for other website specific data (Site Preferences)
• sessionstore.jsonlz4 for open tabs and pinned tabs (see also the sessionstore-backups folder)