You can ice fish in flowing water in most parts of the province there is a discussion in the link provided.
http://www.outdoorsmenforum.ca/showthread.php?t=9168
I have ice fished the athabasca (under permit) to collect scientific samples in several areas (check to see if it is open for angling). Backwaters and pool areas definately congregated mountain whitefish and pike.
Never had any issues in those areas with ice conditions but still used extensive safety equipment including harnesses, safety ropes and observers on shore. Would hate to go down and end up trapped under the ice.
It doesn't take much current to end up being swept away if you were to go through.