Have you by any chance already unlocked it and just never actually set it yet? It might be you unlocked it already without even realizing it. The quest to start it begins at Quiver's Hold, the ARC's Guild in New Gridania, specifically at the reception desk in the entrance.

Like unlocking all the classes they'll ask you if you want to be an ARC, say yes and the actual quest will begin, and you'll then have to go and speak to the Guildmistress in the back room. So it's possible you might have already either unlocked the class fully or at least started it already and just forgotten where you have to go.

EDIT: Oh no problem, I'm glad you found it easily. Good luck with the way of the bow!