Koe scored two points in the fifth end to make it a 3-2 game, but Martin responded with three points in the sixth to pull away with the victory.
Martin scored three singles in the first three ends. Koe added another point in the seventh.
In other quarter-final action, Glenn Howard of Penetanguishene, Ont., scored three points in the fifth end to edge Sweden's Niklas Edin 8-4.
Winnipeg's Jeff Stoughton put up two points in the fifth to break a tie, and added two more in the seventh for a 6-4 victory over David Murdoch of Scotland.
China's Rui Liu downed Denmark's Rasmus Stjerne 6-3.
On the women's side, Ottawa's Rachel Homan won twice, first beating Winnipeg's Chelsea Carey 5-1 in a tiebreaker, and then bumped off Heather Nedohin of Sherwood Park, Alta., 6-3, to advance to the semifinals.
Winnipeg's Jennifer Jones also moved on to the semis by edging Stefanie Lawton of Saskatoon, 6-5.
Scotland's Eve Muirhead was a 5-4 winner over Margaretha Sigfridsson of Sweden, and Mirjam Ott of Switzerland got by South Korea's Ji-Sun Kim 3-2.