This is a bit of a trick question. Technically, according to the story, Nick has a girlfriend back in his hometown. He is already dating someone when the book begins.
As the story progresses, the reader gets the sense that Nick has begun dating Jordan Baker, one of the more minor characters in the book. Jordan is a famous woman golfer. She is also not the most reliable of characters as she is very self-centered and has a thing for bending the truth and gossiping. It is even generally accepted that she cheated to win her first major tournament.
Dating may or may not be a stretch to describe their relationship. It isn't discussed in much depth in the book. Definitely they have some sort of minor relationship, but just how "romantic" it is in nature is a bit debatable. By the end of the book whatever they may have had has dissolved.