Some jewelry stores will purchase your diamond, however they only want to purchase your diamond at a very low price. They can buy from a wholesaler any time they want with one telephone call. Often a wholesaler will actually lend them stock for free in hopes that a customer buys it, only then will the wholesaler get paid (a process called memo in the jewelry industry).
A jewelry store will only pay cash when the price is much less than they can purchase from a wholesaler, and only if they need the type of diamond you are selling. It may be too big, too small, wrong color, wrong shape or simply lower quality than the goods they normally sell.
They may offer to take your diamond on consignment, and you will get paid when - and if - the diamond sells. Unfortunatly, that may take months or years - as the demand for a used diamond (estate jewelry) is not as high as new jewelry. They end up getting free inventory, much like they get from a wholesaler, and you end up waiting for your cash.