Setting up Google Apps for Business account as a Mailbox in FogBugz can be done similarly to any other email provider.
If FogBugz does not send or pull emails from your Google Account, this might be because you are not using App Passwords.
Without 2-step-verification (without App Password)
Setting up a New Mailbox can be done in the same way for Google accounts as for any other email provider.
With 2-step-verification (with App Password)
In case you use a Google Apps for Business account that has 2-step-verification enabled you will need to use an App Password that you have to configure as described in the Google App Passwords setup page and then put that password in the Password field of your FogBugz mailbox.
This method works for both POP3 and IMAP accounts.
If you use the same mailbox also for sending emails, update the SMTP password under the Site Configuration -> Mail to use the same App Password you just generated.