Rather you should really do this instead. What you talking is sync and will not work once the other original computer is gone so will all the data will go with it. Sync isn't a backup for this.

https://support.mozilla.org/en-US/kb/back-and-restore-information-firefox-profiles https://support.mozilla.org/en-US/kb/recovering-important-data-from-an-old-profile https://support.mozilla.org/en-US/questions/966133

You can copy certain files with Firefox closed from the old device to the new device to transfer or recover personal data. Note that best is to avoid copying a full profile folder.

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;32+) and key4.db (decryption key;58+) for Passwords saved in the Password Manager
• cert9.db (58+) for (intermediate) 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)

• https://support.mozilla.org/en-US/kb/recovering-important-data-from-an-old-profile