Best to check well in advance with the Canadian authorities about the necessary paperwork. If you do not have your permits in order, you risk getting the gun confiscated, and prosecution.
American's crossing into Canada and neglecting to declare their firearms makes the news every now & again ... best case it seems like you are looking at a $10,000 fine and some time in jail.
This US Embassy page seems to have a good summary of what you need to do to bring firearms into Canada. Of note, they say the following under the heading "Transport through Canada to/from Alaska":
The easiest way to transport firearms is to have any firearm(s) crated and shipped to your destination via commercial carrier.