While FogBugz Mailboxes do not work with multi-factor authentication, in the case of Google Apps for Business you may use an Application Specific Password (ASP) instead. Simply configure your ASP as described here, and then put that password in the “password” field of your FogBugz Mailbox.
Using FogBugz On Demand? We've recently rolled out a new sidebar as part of taking FogBugz forward. Please see this article for details on what's new, what's changed, and where you can find all your favorite things.