"Occupy"? I think you might be thinking of Iceland, as Greenland isn't usually considered "occupied", even though the US played an important role there during the war. Anyways though, Greenland played an important role, specifically in regards to weather! I wrote up a piece some time ago about the "Weather War" fought in Greenland, which I'll repost here:
By far one of the least heralded campaigns of World War II was the hunt for Axis weather stations set up in remote parts of Greenland. The United States actually began doing this in 1940 at the behest of the Government following the German invasion of the country. The job fell principally on the shoulders of the Coast Guard at that point, who patrolled with ships and aircraft, looking for German weather ships, or supply boats attempting to reach weather stations the Germans had set up. They were also assisted at this point by native Eskimo trackers who assisted in spotting.
The reason Greenland was so important in this regard was that a weather station set up on Greenland's eastern coast - which is immense - offers an excellent window into the weather fronts as they move towards Northern Europe. Obviously weather plays a huge part in military planning, and this being before satellites allowed such easy predictions to be made, the extra day of forewarning offered by a station in Greenland was super important! So Germany wanted to set them up there, and it fell to the United States to protect Danish interests in not having this happen. The first direct combat between Germans and Americans (and by direct I exclude convoy contact with U-Boats) occurred during one of these patrols when a Coast Guard cutter, the USS Northland, boarded and captured the Norwegian flagged ship Buskoe. A landing party went ashore and captured three German soldiers operating the weather station the ship had been resupplying. This all happening three months before America entered the war!
Aside from the Coasties though, the "Sledge Patrol" - a 15 man, mixed force of Norwegians, Danes and Eskimos, all supported by the US - spent much of the war patrolling the coast hunting Germans as well. Only, doing it on land in subzero weather instead of in a comparatively warm boat. On dog sleds, 2 and 3 man patrols would head out for a few months and attempt to find German weather stations in a cat and mouse game. Generally, the Germans were the mice and had to pack up their stuff and flee if discovered (the units obviously were to small to take on the Germans directly, so could merely radio in the position. The Germans had at least four separate weather teams on Greenland that I can find mention of, and the teams would number over a dozen in some cases.), but the Germans did strike back and attack the Sledge Patrol's base-camp at Eskimonaes, killing one member of the team, Eli Knudsen (the only loss they endured).
The last land-based weather station of the Germans was knocked out in October of 1944. Based on Little Koldeway island, the German station was spotted by the USS Eastwind during a patrol. A landing party of Coast Guard sailors (Who, as part of this role, underwent special training under the supervision of commandos), made a nighttime landing and caught the Germans by total surprise, and were able to get most of their documents intact even! No more land-based stations were attempted after that, although off-shore trawlers were still utilized (The Eastwind would take the Externsteine as a prize only a week after the raid on Koldeway).
Thats pretty much the bulk of the fighting that occurred. Nothing in my book makes any mention of plans for anything beyond that limited military presence by the Germans, and if you take into account the location of Greenland relative to where Germany would launch any sort of invasion force, you can see how mind-bogglingly impossible that operation would be. Crossing the comparative puddle known as the English Channel was ruled out as too tough to undertake, so the arduous journey needed for a proper invasion and occupation of Greenland would be neigh impossible.
The source I'm mostly drawing from is "War in the Outposts" by Simon Rigge. Great book from Time-Life.
I've never read it, but I've heard this book is a great account.
More importantly it prevented Britain from having to do it, with their stretched resources. And upheld the Monroe Doctrine and previous statements regarding Greenland that the US would oppose any 3rd nation trying to establish themselves in the Western hemisphere.
It also became an important stop over for aircraft being flown to Europe. And there were in act German installations on the East Coast for 2 years! They were weather monitoring stations which provided important intel to the U-Boats and German High Command. So yes it is realistic to say that Greenland was under a threat of being invaded, the Germans actually set up shop in several locations for 2 years.