You have to ask the Licensing Office in Québec. Generally, you have to change your driver's licence within 60 to 90 days when you move to a new province (or a US state) but the exact time limit depends on the laws of the province. In Québec, the driver's licence fee includes part of the insurance and therefore it is a bit more expensive than that of Ontario. However, the overall insurance premium (there is another part that you need to buy from a private insurance agent) is lower than that of Ontario.
If you do not plan to drive, you do not have to change your Ontario licence before it expires. However, if you do not change it before it expires, you may need to retake the driving exam in Québec all over again.