There are numerous sites that connect to Google account services:
For example:
My proof-of-concept will be written in Python locally. When it works, I intend to upload it to Google App Engine.
How can I connect my own app or website to Google Account? Links to Official documentation, License documentation and tutorials are highly appreciated.
It really depends what you want to do with that Google Account. If want to authenticate users, look at their OpenID documentation, if you want to integrate with data from that account, look at there data APIs and get an OAuth consumer key/secret.
For Gmail data, as far I ca know, there's no native way to do it from AppEngine (you can't do network sockets as an IMAP connection would require). One alterntive is the product we built: http://context.io