Is it okay to give a grown cat milk?

Yes, Go ahead. Although certain chemicals do build up in the cats digestive system which could be harmful to its health. ie: urinary tract

ADD:

Please be careful giving a cat milk. Most cats are lactose intolerant and will become sick on even a small amount of milk. The safest option, if you want to give your cat milk, is to purchase some specially formulated cat or kitten milk which can be found in pet isles in most stores and pet stores.