This is slightly story-related, but it has no bearing to story elements. Introduced in the Museum of Vanity, Treasure Demons hide within the breakable treasure objects you can find throughout the Palaces (or in the case of Mementos, just casually running around). From this Palace on, you can expect to find them mixed in with the treasure environments. If you Down one, you will have the chance for an immediate recruitment into your party. This will allow you to use it as fusion fodder for special Persona. If you kill it, you'll get a lot of money and experience.
A suggestion by PhillyCh3zSt3ak offers you to try for the second TD in the second Palace, which has a prominent weakness against Nuclear.
*Do note, the first two Treasure Demons you come across are a bit easier to down. In the fourth Palace, the Stone of Scone TD will require a Curse-dmg ability (which I personally didn't have).
I'd suggest nabbing the first one you come across as a recruit, so you can register it to your Persona Compendium.