The Quick Slots are a set of 10 slots for items that can be used/consumed during battle.
These are typically stocked with scrolls and potions that are needed during battles.
During battle, the items in the Quick Slots can be accessed using the keyboard's function keys (F1 - F10) during your turn.
Outside of battle, the Quick Slots are part of the Inventory window and can be accessed below the Inventory window outside of battle. If you have and are displaying Extra Inventory Bags, the Quick Slots will be below those. Outside of battle, you can refill or switch out what is in the Quick Slots just like any other Inventory slot.