This is really three problems, as follows:

(1) Accessing the Google Contacts API to retrieve someone’s details based on their phone number.
(2) Passing the incoming caller’s number to an AGI script.
(3) Displaying the details retrieved from Google on your screen.

Presuming you already know how to write a program to look up a Google contact’s details from their phone number, you just need to turn that into an AGI script. Then, in your dialplan, pass the incoming number to that script.

If you want Asterisk itself to have anything to do with the Google data, you will have to return them by setting channel variables within the script. Or if you are going to use some external means to pass the data to the user, then you can have your script fork itself, detach and return straight away.