pawnshop which is a icon like a stack of books, there is a sell tab on the when you ask to buy something

Easiest way to sell them is to bring them back to your apartment and sell them using the item tab of the decor thing. The more RP and legitimate way of selling them is to go to a pawn shop.

Note that you can get a much better price for diamonds from the Black Market dealer, but you first have to find the correct password for them to service you.