It is not normally possible to get back deleted passwords.But there are three possibilities.

1. you have them on a 2nd device
2. using somesort of file recovery system. Or existing backup files
3. possibly using sync

Expanding on the initial comment above.

1. 2nd Device
   Sync is not intended that the server be a backup, it is intended to synchronise with a 2nd device. Do you have a 2nd device and does that have your passwords on it ?
   If so you can copy passwords from that. If there are not many copy manually, otherwise it will need a file copy (There are two files to copy the encrypted paswords and the key. I will explain if required)
2. File Recovery
   May just be possible. Best if you have a deliberate pre-existing backup of the profile. Third party file recovery software may help but need a large helping of luck and quick action. Windows has a previous version system that could help. I do not know about Macs (? TimeMachine ?).
3. Sync
   Backup whatever you have now on both devices by backing up the profile. If you synced while you had passwords, or the last sync was when you had passwords then syncing should recover them. If you have the profiles backed-up you do not really have anything to loose by trying.
   • Back up and restore information in Firefox profiles

If you are using the Time Machine on your computer then you can see if you can restore the signons.sqlite and key3.db files.

• http://www.apple.com/support/timemachine/

• https://support.mozilla.org/kb/Recovering+important+data+from+an+old+profile

You can use this button to go to the Firefox profile folder:

• Help > Troubleshooting Information > Profile Directory: Show Folder (Linux: Open Directory; Mac: Show in Finder)
• http://kb.mozillazine.org/Profile_folder_-_Firefox