How man NCAA mens basketball championships has UNC won?

No. In fact, they have never met in any round of the NCAA Tournament. The closest this came to happening was when both teams reached the Final Four in 1991. Duke upset defending champ, heavily-favored and previously undefeated UNLV to earn the berth in the finals (avenging a humiliating loss in the finals the previous year); however, UNC was upended by Kansas (with UNC alum, former Tar Heels assistant and future head coach Roy Williams at the helm). Duke went on to defeat the Jayhawks for the championship.