Hi Marvin,
I think it should be possible in the right year at the right time of year, at least in theory. The Qu'Appelle would be fed from Lake Diefenbaker at the dam near Douglas Provincial Park. I expect you'd find some beaver dams along the way, but, in a wet spring there should be enough water to float a boat without much trouble and would probably paddle over such obstacles. Maybe lots of snow in the mountains to fill up Lake Diefenbaker would help too. I have no first-hand knowledge, but am quite happy to speculate.

Have you tried contacting paddlers in the southern part of the province? How about Geoff Horn at Fresh Air Experience in Regina?
Cheers,
Bryan