At about 2040 eastern time, a Mooney M20K suffered an apparent nose gear problem and was damaged during landing at Greenwood Lake Airport. The pilot received minor injuries. The pilot said he had transported another pilot to Stewart International Airport to pick up another airplane. At the landing at Stewart, he detected a nosewheel shimmy. The two pilots examined the nose gear but could find no anomalies. The two then departed in separate airplanes for the return flight. The pilot said that, after he touched down, the nose of the airplane veered to the right. He pulled back on the yoke to ease the nose up, but he pulled back too hard and the airplane became airborne again. He thought he would touch down on the runway again, but instead, the airplane landed between the runway and taxiway and struck rocks.