I connect firefox browser with my gmail account. The connected gmail account is not syncing old and new mails properly for single connected account
Other gmails accounts are working correctly. The only gmail account connected with firefox not syncing mails, contacts and all
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I can share registered email if needed
That sounds like a support issue for Gmail support; not a Firefox support issue as Firefox doesn't "connect to Gmail" anymore than Firefox "connects" to any other webpage. Any Gmail feature beyond the display of email content is done thru the Gmail software; and possibly thru a Gmail add-on for Firefox, support for same would be rendered by the developer of that add-on not here.
If you are having an issue with one Gmail account viewed thru Firefox, try clearing the cookies in Firefox for that Gmail account and see if that helps.