I'm in the Level 12 (Gladiator - Ul'dah) and my Hunting Log isn't enabled. What quest (main or not) it's necessary to complete to unlock the Hunting Log?
According to the Final Fantasy XIV wiki:
Hunting Logs, which classes gain access to after their first class quest, are a series of one-time hunts that provide a bonus amount of experience points after defeating a certain number of enemies.
So you'll need to do your class quests in order to unlock it. It doesn't matter what level you are; it's all about starting your path down your chosen class.