This is a 5-year-old bug, judging from the date on the wowhead post with the most upvotes mentioning this exact problem, which I have no recollection of running into before. Rather than necrotize a post here from several months ago, I am posting it again.
I logged in to work on the Draenor Pet Brawler achievement and decided I would just use the Garrison pet battle arena to do so. Did the battle twice, then figured I would pick up the Mastering the Menagerie quest from Lio the Lioness. Picked up the quest, then went and completed the battle. To my surprise, I did not receive credit for completing the quest even though I won the battle.
Yes, I know I can log into another toon and complete the quest, but that seems to be a shoddy workaround.