The 22-year-old Brown has 18 goals and 11 assists in 51 games for Portland this season.
Brown was a second-round draft pick of the Coyotes in 2009. He didn't sign with Phoenix until last year after he finished his college career at Michigan, where he had 34 goals and 46 assists in 125 games.
Klinkhammer leads the Pirates this season with 44 points in 52 games. He's tied for eighth in the AHL in points and leads Portland with 30 assists while ranking third on the team with 14 goals.
The 26-year-old Klinkhammer was signed by the Coyotes as a free agent last July after NHL stops in Chicago and Ottawa.