Is there a fast way to unlock Torment mode?

You only need one of your characters reach Level 60 to unlock Torment difficulty.

Here's the game guide.

How do you unlock Torment?

If you reach level 60 with one of your character you unlock Torment difficulty. See here: Game Guide - Diablo III

You’ll unlock Master difficulty when any of your characters completes the game, and Torment difficulty when any of your characters reaches level 60.

Is there a fast way to unlock Torment mode?

Yeah, fast leveling! Ask a friend or ingame, if he can help you leveling. Make a game on highest difficulty and let your friend kill everything. If you killed Malthael you unlock Adventure Mode. Do Bounties on highest difficulty and earn even more experience.