Yes and no. No, there is not and never has been any romantic liaison between them. But it's fair enough to assume they are good friends who have fought against the same evil for many years. It was Galadriel who first summoned the White Council (one of the meetings of which we see in The Hobbit) and she would have wanted Gandalf to be the leader of the council instead of Saruman to whom the task was appointed in the end. It was also Galadriel who sent one of the eagles to seek Gandalf after he had fallen with the balrog in the Lord of the Rings, and it was she who dressed him in white after his return. They are both also Ring-bearers, bearing one of the three Elven rings each. Hers is the ring of Water (Nenya), his is the ring of Fire (Narya). These Rings are different from the One Ring. Nenya, Narya, and Vilya are benevolent. They do not corrupt the user.
So, for sure there is a connection there - a platonic one between two of the wisest beings in Middle-Earth during that time. I don't think the movie necessarily implies there being anything more either, but that is of course left for the imagination of the viewer.