Quote Originally Posted by Bixillarla View Post
It is a main story quest so you can't miss it.
No, it's not a main story quest. You have to reach a certain point in the main story before it's available, but the "My Little Chocobo" quest itself is a sidequest. (I'm guessing this is because it requires accumulating 2,000 Grand Company seals, which takes a while at that level, and they didn't want to halt the story there for that long.)

The main story will have you join one of the three Grand Companies, Twin Adder, Maelstrom, or the Immortal Flames. Once you've done so, then you can take the "My Little Chocobo" quest available at the GC headquarters. Completing that will give you a chocobo and unlock the ability to summon mounts.