Suzaku was added in patch 4.4 of the Stormblood expansion.
Suzaku becomes available at level 70 as part of the “The Fire-bird Down Below” quest, which is given by Bunchin – The Ruby Sea (X:27, Y:16).
The trial has a minimum ilevel of 355 to enter and rewards you with:
Allagan Tomestone of Poetics: 15