Provided you intend to live in Quebec and you tried to live in Quebec, you could live anywhere else.
You could establish yourself and later go to another province, because the day you become permanent resident, you are protected by article 2 of the Canadian constitution that guarantees freedom of movement. This being said, during the process to get PR you should maintain the intention to settle in Quebec, and then give a legitimate chance of establishing yourself there. The above being said, no worries: The climate seems worse than it actually is, and with proper guidance from your representative from Quebec, you would find it much easier to adapt than you may think.
Canadian permanent residency allows you to live in any province. Once you have your PR card, you can live where you want.