How do I sync contacts with Gmail on an iPhone 5?
This Google help document shows the normal setup steps, as well as some troubleshooting steps. 

For troubleshooting, ensure that Contacts is turned on, and then click on the account information above the list of mail, contacts, calendar and notes. This will show your Gmail account information. 

Click on advanced, and ensure that SSL is enabled. 

It may be helpful to create a new contact locally on the device and see if it syncs up to Gmail.