From what system/FF version to what system/FF version?
If these are different, the files may not be compatible.

You could try this:

https://support.mozilla.org/en-US/kb/how-do-i-set-sync-my-computer
Sync your bookmarks, history, passwords, add-ons and open tabs with another copy of Firefox. Learn how to set up Sync.

https://support.mozilla.org/en-US/kb/how-do-i-choose-what-types-information-sync-firefox
We'll show you how to choose which types of information (bookmarks, history, add-ons, and passwords) to share across all your devices through Firefox Sync.

https://support.mozilla.org/en-US/kb/firefox-sync-troubleshooting-and-tips

https://support.mozilla.org/en-US/kb/sync-firefox-bookmarks-and-browsing-history-iOS
Sync your passwords, history, tabs and other browsing information across your iOS, Android and desktop computers with Firefox Accounts.

What Firefox version was installed on the old computer ?

You would normally copy places.sqlite to transfer bookmarks and history.

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You can copy files like these with Firefox closed to the current profile folder to recover specific 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 (passwords) and key4.db (58+) or key3.db (57 and older) (decryption key) for Passwords saved in the Password Manager
  if you only have key3.db then make sure to remove an existing key4.db
• cert9.db (58+) or cert8.db (57 and older) for (intermediate) certificates stored in the Certificate Manager
  if you only have cert8.db then make sure to remove an existing cert9.db
• persdict.dat for words you added to the spelling 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)