Let's be clear. It's up to the DM.
Al Sun cites the "downtime activity" rules, which applies specifically to player characters.
You can spend time between adventures learning a new language or training with a set of tools. Your DM might allow additional training options.
Nothing in that section refers to anyone but you, the player-character, receiving training. There are no general rules regarding animal training, let alone specific rules for this particular situation.
As players, you may think it would be quite cool to have flying mounts for your characters. However, your DM is likely focused on something more important than "cool". His job is to provide an adventure milieu with interesting stories and exciting challenges. Flying mounts can ruin a lot of stories and make some challenges trivial. If your DM rules that the mounts cannot be retrained or your characters lack the saddle-skills to ride them, understand that he's simply trying to give you the best overall experience. Frodo might have thought it cool to just ride an eagle to Mount Doom, but it would have made for a boring book.