Technically speaking you get the pets as a reward for honor level 50. Prior to patch 7.1 you did not need to prestige to get them. Since patch 7.1 you automatically prestige once you reach the required 50k honor for each rank.
Brian Holinka (PvP Dev) stated during a BlizzCon panel that there won't be a prestige rank reset when Legion Season 2 starts and that they want to add more rewards for the new prestige ranks. Like Peanutty said this implies that the current pair of pets will be available till the end of Legion.
Rumor on wowhead.com and other WoW related forums is that the new prestige pets will most likely be rewarded for reaching prestige rank 4 & honor level 50. This would somewhat mirror Season 1 where you start with the pet (PR 1), get your PvP weapon skin (PR 2), something I don't have yet (PR 3) and a new mount (PR 4).
To be sure that you don't miss the pets start doing the PvP tower WQs (300 honor) as often as possible on a character that owns a guild banner. If you use the banner while completing the tower, you will also get the honor bonus (5%, 10% or 15%). To further increase honor gains do this grind on a character that can auto-complete WQs from the Tier 5 talent of your Order Hall (e.g. Demon Hunter, Paladin, etc.). Use this feature on cooldown (18h) for the FFA WQs that reward 500 honor each. Your guild banner will work here as well, unlike doing the actual quest in the world.