The major conflict that emerged between the former allies was the Cold War. It was really starting to emerge even toward the end of World War II.
The problem was that the Soviet Union and the Western Allies did not trust each other at all. The Soviets thought the West was out to encircle and destroy them. The West thought that the Soviets were going to try to dominate the world. This led to conflict very soon after WWII ended.
Later on, there would be conflict between the three Western allies, but this would not be until quite a bit later.