Neglecting the fact that collecting Qatar miles may not be your best option to fly with Qatar/OneWorld, the best option you have, living in Canada, is probably to collect Starpoints (from Starwood Preferred Guest loyalty progam)
Starpoints can be transferred at 20k:25k (1:1.25) to Qatar, and there are options in Canada to acquire Starpoints without actually staying with SPG.
- Signing up for SPG credit card by American Express
- Signing up for any credit card by American Express that earns Membership Rewards points, which can be transferred to Starpoints at 2:1 (in Canada)
(Canadian) SPG credit card typically has a sign-up bonus of 20-25k Starpoints, and the Membership Rewards points earning cards has a larger range, around 25-50k MR points.