As Souvy observed, PRs have mobility rights too. The limitation, such as there is one, is that Quebec applicants have to sign a declaration that they intend to live in that province.

So it all comes down to intent. Applying through Quebec as a short cut to another province isn't allowed. Changing your mind once you get there is OK.

Of course, leaving Quebec very soon after arrival might lead a reasonable person to wonder whether you ever really intended to settle. In that situation, it might be helpful if you could show that the move was a reaction to some external circumstance rather than the plan all along.

There's a good chance you'll never have to prove anything to anyone. I suppose someone at CIC might notice the move when you later apply for a PR card renewal or for citizenship.

Think it's worth mentioning that you're easily trackable on an annual basis when you file your taxes. If you land in Quebec with a CSQ and then move straight away, Quebec won't have a record of you filing your provincial taxes.