It depends on what you mean by "notify ATC."
As grizz pointed out, there is no requirement to talk to ATC when enroute as they won't provide you any services in Class G.
However, in order to operate IFR you need to have filed a Flight Plan or Flight Itinerary. This is still done by calling or electronically filing, the plan or itinerary, so there is still a notification process.
In the case of a flight plan, you'll need to obtain the clearance and this is normally done through the radio operator in Hall Beach or Pond Inlet. These are the same people you'd file the arrival report with, when they are there. Otherwise, this all has to be coordianted through Arctic Radio (or whatever they're calling it nowadays) either by phone or on 126.7. But, because none of the services are provided by ATC (except a more closely watched alerting service) below FL230 in Northern Domestic Airspace, most pilots not flying for one of the airlines will simply file an itinerary and avoid picking up a clearance alltogether. An IFR Flight Itinerary is certainly the way to go to make your life easier.
You'd have to remember too that almost all of your flight will be conducted in the Canadian ADIZ (which recieved a major overhaul in 2018). The boys and girls at NORAD don't bother you too much until you're way off track or far too early or late from what you filed, so updating your flight planned timing and routing should be given some consideration to avoid egg on your face!