If you wish to apply for a study permit while visiting Canada, your study permit application will be processed as an "outside of Canada" application through your home country's visa office.
Generally speaking, it is better to apply for a study permit from outside of Canada and not as a visitor in Canada. If you are already in Canada, it is more difficult to show ties to your home country. However you can certainly apply while you are in Canada as a visitor. Expect processing to take a number of months. You do not have enough time to get a study permit approved for the January semester. Make sure you keep your visitor status valid while you wait. This may mean applying to extend your status as a visitor in Canada.
