Dear Mahesswar Shri Mohanty,
Good day to you and thanks for reaching out to Microsoft Community.
As per the description you shared, I have a general understanding of your query, regarding the problem of Gmail account which is not syncing on Outlook for Mac. In general, this might be due to the Gmail settings, as Outlook need access to your folders in Gmail, so make sure that the folders are set to sync in IMAP. Once you've enabled IMAP in Gmail and set your Gmail folders to sync in IMAP, refresh Outlook. Or you can remove and readd account in Outlook for Mac after checking your Gmail settings and see if it makes any difference.
Kindly note thatit can take up to 15 minutes for syncing to refresh. After removing a Google account from Outlook for Mac using “Remove from all devices,” adding the same account back may take up to 24 hours.
See Fix problems connecting your Google account to the Microsoft Cloud - Microsoft Support.
Meanwhile, we understand how frustrating it is and sorry for your experience. If issue persists, since it only happened on the Outlook desktop side, but not server side, unlike Office 365 Exchange server which we could ask the related team to check on the backend for you, in this situation, we highly recommend you contact Outlook for Mac technical support, and they would be able to figure out the issue by collecting diagnostic data and checking on Outlook desktop app side. See Contact support within Outlook for Mac - Microsoft Support and contact them at your convenience. This would be the most efficient way in handling this case for you.
Thanks in advance for your understanding! Your patience and cooperation will be highly appreciated. Hope you all the best!
Sincerely,
Tammy | Microsoft Community Moderator.