Labrador was originally part of New France and thus historically part of what has now become Quebec.
What is now Quebec technically never relinquished its possession of Labrador and its ownership was superseded by the conquest by the English crown.
Quote "After the the English conquest of Canada, Labrador was placed under the jurisdiction of Newfoundland. In 1774 it reverted to Canada, only to be returned in 1809. Shortly after the Conquest, English merchants and adventurers began to occupy the coast, and within a few decades were joined by French-Canadian and Jersey interests as well."
Thus Newfoundland was given "caretaker" status over Labrador several times by the Brit's.
Quebec does not recognize the loss of Labrador as part of it's defeat at the hands of the Brits.
Newfoundland bases its ownership on historical control and possession.
Quebec still has trouble accepting that.
As to the border between Labrador and Quebec: Lone Wolf is right.
Historically various watersheds and their direction of flow were agreed to define the area's of Quebec and Labrador.
At the time all concerned parties agreed to this system of division.
At the time it was an extremely common system.
Nothing has changed and it is obviously quite complicated to map numerous entire watershed's.
In fact they tend to change over time.
Thus there is no grand conspiracy, no complex plot.
Just a fairly involved process for mapmakers, cartographers and lawyers to slog through the geomantics of the region and then make there professional findings known.