June is the month for trout. This fish swims close to the coastline, and can be caught from points and piers. However, from the middle of July to the end of September they enter the rivers and it is a paradise for river fishing.
In Narsarsuaq, you can fish directly in the fjord, or hike to rivers. Another option is to make an agreement with local outfitters, who will provide transport to the rivers that are farther away. There are also good opportunities for fishing with dry flies by the upper waterfalls.
As a tourist, you do have to buy a fishing license when fishing in Greenland. It costs about DKK 250 for one week (EUR 34) and must be paid at the local post office.