In Palom's Challenge Dungeon, the Gil Bird is only encountered on the Mountain Floor, runs away fast and does not fight. The player must act fast in order to kill it. Have Leonora cast Slow while Palom uses Bio, and after the first round, the Particle Bomb Band should be enough to finish the Bird off, or at least weaken it considerably. Normally it only leaves 999 gil, but when fought during a Full Moon they yield 9,999 gil.
In Edward's Challenge Dungeon, the Gil Bird appears as a miniboss in one of four locations, depending on the Moon Phase. Defeating it yields the Gil Bird Egg, needed to sell to a merchant in the dungeon for a substantial amount of gil.