You can enable the standard logs in the Preferences -> Advanced:
Enable the “Debug mode logging” for the verbose version of the logs. For the most verbose level of the logs you have to enable the IMAP transactions logs but be careful: it can quickly fill your memory (it creates a lot of data).
To See the standard Logs
Open the Utility -> Logs window, they will browse in that window:
Before closing it remember to disable the Logs in Preferences otherwise it will go ahead to store data in that box.
To See the Console Errors
Open it (Application -> Utility -> Console) and look for the BackUp Gmail lines: