I just joined yesterday and well... i put up my Roc to this quest:

'Guard College is looking for a Flying Animal to take part in the riding lessons'

So um.. How can i compleate this quest?

It takes 4 hours to finish. Once you hit the submit button, the timer starts counting down on its own and after 4 hours, your pet returns with the quest reward (usually just money, but sometimes it bring items too). 

You can see how much longer the pet will be busy by checking on its profile. On the right side, there's a box that says "Effects". In this box there will be a short statement telling you what the pet is busy with and how much longer it will be. This is the same if a female is pregnant and it also tells you when a baby will grow up and when a captured pet will be tamed. 

Alright.
Thank you for answering.