Seems like someone forgot to patch this…
Giant kills (as well as other monster kill count quests) only count if you are within 10 levels of the mob you kill. So if you were level 60, for example, the progress for the quest is only counted if you kill mobs/giants that are level 50 or higher. Mountain Mages are level 21, so if you are between level 11 and 31, then this is an issue, but otherwise it is intentional that it does not count for the quest.
Oh so that’s why I think devs forgot to patch this. I was at Lv60 the time I post this. So if I kill the giant Exalted Samurai, does that count, and what if it is lured?