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All but one of the "Capture x monsters" quests (I don't remember exactly which one, maybe Savage Bebe?) require you to use a special weapon during the fight in order to get the quest item. That special weapon will be given to you after you accept the quest and it looks like a pink rubber mallet (the same one used in the typical "hit the hedgehog" game, don't know what it's called officially lol). You don't need to use it during the entire fight. I'm a magic class so I usually weakens it with my regular weapon then swap for the kill.
yes, it is something the quest doesn't even mention. When you start the quest, you are given a 2-Line Capture Mallet, I believe it was. That must deal the final blow in order for the item to drop. It is the ONLY quest I found where you capture something that has a special way to get the item it drops. The quest, I believe calls the Green Peco Peco Bebe, but they are actually Green Peco Peco Baby.