The most recent three-home run game I can recall in the organization was Richie Shaffer kind of early after his promotion to Durham in 2015. If there was a minor league play index, I could know for sure.
Triple-A Durham Bulls (34-42)
Durham scored first, but it could not add on in its 4-1 loss to Lehigh Valley. Eury Perez, who apparently plays for the Durham Bulls, led off the game with a double and scored on Richie Shaffer's single later in the inning, but the Bulls would only get one hit over the final six innings.