New Year's Day was a human holiday celebrated annually on January 1, commemorating the beginning of a new calendar year. These celebrations were normally combined with those of New Year's Eve, celebrating the end of the previous year on December 31. One of the largest traditional celebrations was held in Times Square in New York City.
Q while visiting New York City, briefly befriended a human woman in anticipation for New Year's of Jan. 1st, 2000. Unbeknownst to everyone else but himself the ball that is dropped to usher in the new millennium was sabotaged by terrorists. It kills all in attendance, except for Q. He then leaves pondering how blissfully ignorant mortals are to their concepts of importance verses the greater universe. (TNG novel: I, Q)
San Francisco celebrated New Year's Day 2378 with a long congratulations to the USS Voyager and its crew, following their recent return from the Delta Quadrant. (VOY episode: "Endgame", VOY novel: Homecoming, SNW short story: "Uninvited Admirals")
In an alternate timeline, Trip Tucker and T'Pol attended the celebration in Times Square for the beginning of New Year 2200. A violent confrontation broke out at that event, and Tucker was tragically killed. (Star Trek: Myriad Universes novel: A Less Perfect Union)