Quota is the space you provider allocates on their server to store your mail. Compacting can free some space still used by previously deleted but unexpunged mail, so might work as a temporary solution.

A long term solution requires you to have less mail stored in your mail account.

Move mail to local folders or delete it and empty the trash to actually do more than move the mail to another space consuming folder.

Thanks for your response. I moved mail to a local folder and also tried deleting it and emptying the trash but I would get the same error.

I suggest you use your providers web mail and go there and empty the trash first. My guess is that nothing in Thunderbird will work until you free at least a little space on the server. Deleting mail consumes space in your account and the mail may not be expunged from the server to free space until you restart Thunderbird anyway.