You first open the achievements tab then go to Collections - Mounts, scroll down a bit till you find the achievement that I think it’s called Lord of Reins at that point, one of those blue achievements. You click it and then you see your progress. If it’s not named that then you can check here https://www.wowhead.com/achievements/character-achievements/collections/mounts
I’m currently at 344 mounts myself, so close to getting the Biting Frosthard Core mount!