The chocobo dungeon is inside the chocobo ranch itself and can be be unlock by:

Quote Originally Posted by GAMEFAQ
1) Capture and dispatch THREE level 1 chocobos to the Calm Lands. Make sure
that all THREE return safely.
2) Raise and dispatch THREE level 2 chocobos to the Calm Lands. Make sure that
all THREE return safely.
3) Raise and dispatch THREE level 3 chocobos to the Calm Lands. Make sure that
all THREE return safely.
4) Raise and dispatch THREE level 4 chocobos to the Calm Lands. Make sure that
all THREE return safely.
5) Raise FOUR level 5 chocobos. Dispatch THREE of those to the Calm Lands and
make the fourth your runner that doesn't get dispatched. Make sure that all
THREE return safely.
6) Leave the ranch. Board the airship. Return to the ranch. Clasko will
approach you.
I have tried this myself and it works. I got the AP egg and the Higher Power GG thanks to this.