If you noticed in the introduction to "Compete Against Others", the 3 other choco guys you're competing against, One of them is named Bashraf.
Bashraf appears in two quest so far that I know of, one is for the Bibiki Seashell(a cameo role), the other is in chocobo raising(he's responsible for the mental attributes of their bird iirc). He does not have a position in either Al Zhabi or Whitegate that you can go visit him at. So to get the the Story of a Happy Chocobo, you probably have to beat their chocobo repeatedly in races, and you'd then receive it as a reward. Probably beat them 3 times is my guess. If so, we'll find out today when more people compete against them to win.
Edit for a question: Anyone with blue or black tip feathers. Did you feed your bird any sort of worms anytime during it's growth cycle or stuck to standard greens, carrots, wildgrass, paste?
XI Wiki

