The very simple answer is that they did not like each other.
An answer that is a bit more in depth is this: the Eagles are a very proud race in Middle Earth, and they do not have specific alliances as they prefer to stay out of things. That being said, in 'The Hobbit,' they live in the northern part of the Misty Mountains, not too far from the High Pass near Rivendell, which puts them near the Goblin-town beneath the mountain. According to the story, the Eagles often attack the Goblins and "stopped whatever wickedness they were doing." The Eagles also came to the rescue of Thorin and company when they were being attacked by the Goblins after escaping their cave. Needless to say, the Goblins were not fond of being set upon by the giant creatures.