You are correct that you have 60 days to change over your Quebec driver's license to an Ontario license.
According to the Ontario Ministry of Transportation (MTO), if you are a new resident in Ontario and have a valid driver's license from another province, state or country, you can use it for up to 60 days after you move to Ontario. If you want to keep driving in Ontario, you must get an Ontario driver's license. You must use your Ontario license when you are issued one.
The MTO also notes that new Ontario residents have 30 days before they have to register their vehicles and get Ontario license plates and vehicle permit. The MTO site tells motorist how to license a car in Ontario that is registered in another jurisdiction and how to exchange your Quebec driver's license to one from Ontario.
As part of the vehicle licensing and registration process in Ontario you must show proof of your Ontario insurance company's name and policy number to get the vehicle plated in Ontario.
Ontario has compulsory automobile insurance. Before you can attach plates to a vehicle or renew your registration the vehicle must be insured. In Ontario, private companies provide insurance coverage.
All vehicles must be insured for third party liability for at least $200,000. This covers you in the event that you injure or kill someone or damage someone's property. Collision insurance to cover damage to your own vehicle is a good idea but not required by law.
Click here to get started with your Ontario, Canada auto insurance quote so that you can be insured, registered and plated within the required 30 day time period.