What is the connection? The medieval meaning of club was big stick used to hit things with, the sort of thing slouching cavemen are usually depicted with. However, if you don't have a club of your own, you can always make one. The simplest way to do this is mentioned in a 1625 passage about people warding off elephants:
Many times they fall upon the elephants which come to feed where they be, and so beate them with their clubbed fists, and pieces of wood, that they will runne roaring away from them.
You can make a club out of anything, even your hair. Ladies in the eighteenth century would often wear their hair in a club, which is to say that it wrapped together into what we'd call a bun on top of their heads. And metaphorically, you can wrap people together to make them one powerful object, like a club.
The sense development here is, to be honest, pretty damned obscure. Maybe the haircut was the bridge or maybe the sense of a group of soldiers formed into one blunt and brutal club. But pretty much everybody agrees that the idea of a club of people is that metaphorically they have come together like the instrument carried by a slouching caveman.
So a golf club and a golf club are, etymologically, one and the same.
Both are now weapons