Nice choice, I'm also! :)
Quarry points can't be found with your lay of the land/prospect abilities, and tend to be near streams (black shroud) and in caves. You use your off hand tool (sledgehammer) for quarrying, and there is no mini-game. Just go up the the sparkle, quarry, and then move to the next spot it pops up nearby until you've used all your gathering attempts for that particular location (like a single normal mining node, you just have to move around a bit).
Personally, I don't find quarrying gives all that great of items compared to normal mining, usually things like quartz, slag, or scale bugs. And since it is very common for there to be mobs that aggro in caves, it tends be time consuming at best (swap classes, kill stuff) or dangerous at worst (if your battle class isn't higher ranked).
So if you are trying to make gil I probably wouldn't bother with quarrying. I do find it a fun little side diversion from time to time though! ^^