There is a chance that you will come across Moon Dust, but the amount depends on several factors: the mode in the room, the presence of a phenomenon in it, as well as your skill level in the Twilight Chamber. The higher is your skill level, the more Moon Dust you will be able to find in the room.
An amulet that increases your chance to find an item will help you gain more Moon Dust.
An amulet that increases your chance to find an item will help you gain more Moon Dust.