Rogue Galaxy Quarry: Ancient Fairy
In order to fight this quarry you'll want to fly to the Rose Nebula. Once you reach the Rose Nebula you'll want to head up to the control deck. At the far end of the control deck you'll find the area to initiate the boss fight (as seen in the screenshot).
The item required to summon this boss is Primeval Beef. If you are missing the item required to summon this boss grind some battles and cross your fingers or go treasure chest hunting on a planet you've 100% the map on. In Rogue Galaxy once you pass a certain chapter the loot table of items that can drop for you ingame is updated.
What I mean is, every item has a chance of dropping off every creature after you've passed that chapter that allows that item to drop. At least this is how it works in my experience. Therefore, if you're missing any items to summon any of the quarry bosses all you need to do is grind a little until you get that item to drop for you. Good luck!