How to Sync iphone contacts to gmail
All google syncing is done through the wireless connection now.  Go to the Settings app, and choose Mail, Contacts, Calendars.

Choose Add Account... and select Gmail from the list.  Follow the instructions to connect to your google account, and then select whether you want to sync more than your contacts.

Tested on my iPhone 3GS with iOS 6.1.3 (6.1.6 just updates a few security issues, no change to the contact/gmail sync)