If LiveAgent seems to not fetch emails from successfully connected Gmail account (from which it has already fetched emails), usually it is caused by changing Gmail login credentials.
The first aid in this situation is to deactivate and reactivate the email account in Configuration > Email > Mail accounts.
It is not recommended to remove the email account and reconnect it, because that would cause LiveAgent to start fetching all emails from the inbox. It would find that most of the emails have been already processed, so there won't be any duplicate tickets, but depending on the number of emails in the inbox it may take significant time to process all the emails history.
