There will be "Chocobo Shop" inside the City which looks like the picture below *note the sign of the shop, this place might possibly be the place to get a Chocobo for yourself.

But you need to get some form of "Permission to Ride on the Chocobo" somehow... Seems to me that the Job Level matters ( maybe you need to get to a certain level to get it ).

When riding on your chocobo, you are able to access "Hole Digging" Option from the action command. If you click on it while having some Githal Green...

The Chocobo will keep on digging on the spot for sometime...

If you are lucky, he will be able to dig out an item. If not, nothing will happen. 1 Githal Green will be used per try.