[Ardamir the Blessed]

The Hobbit, 'An Unexpected Party':

He [Bandobras Took (Bullroarer)] charged the ranks of the goblins of Mount Gram in the Battle of the Green Fields, and knocked their king Golfimbul's head clean off with a wooden club. It sailed a hundred yards through the air and went down a rabbit hole, and in this way the battle was won and the game of Golf invented at the same moment.

So if we are to trust The Hobbit, Hobbits probably started playing a type of golf after this. But if we imagine that Tolkien never had written The Hobbit, I doubt that golf would have been mentioned in LotR or any other text instead. Golfimbul's name is the origin of the name of the sport, but in that case 'Golfimbul' had better be a modern translation of this orc-leader's real name in Westron, Black Speech, Orcish or any other language that existed in the North-West of Middle-earth in the Third Age, and that the real Hobbit-Westron name for golf was not 'golf' but something else; I am not sure if it sounds like such a translation (similar to Frodo, Bilbo etc.). To me it sounds more like true Westron (though I am no language expert). Though maybe it works if we suppose that the name 'golf' that exists in many languages today was derived directly from 'Golfimbul' and survived to our days, even though 'Golfimbul' was perhaps true Westron.