I don't recall seeing them in the books, but I might have missed something- are there genies in Rowling's Wizarding World?
Genies receive a brief mention in an article on Pottermore, but they don't appear to be the wish-granting kind.
The ceremony started in gentle style, with river Genies from the Ivory Coast dancing in formation over the surface of the lake. It was only when the Fijian and Norwegian mascots were released that disaster struck.
I've also searched every single piece of canon material, JKR's twitter, the old Potter website, old Pottermore website and current Wizarding World website for any use of the word Genie, Jinn, Djinn, Jinni, Jinnee and Ifrit, but there are no further mentions.