iPhone/ iPad: Sync to Gmail contacts using google exchange service

The default email connection made by the iPhone and iPad only allows access to Mail, Calendar (and Notes on the iPad)

Connection to contacts can be acheived through adding an exchange account connection for contacts only.
See: Set Up Your Apple Device for Google Sync 

Note the exchange server connection offers mail and calendar, but the exchange server connection is unstable (i.e. mail and calendar are subject to periodic drop outs for hours on end) so ONLY choose the contacts option: