Cuba
The words "immediatly following" World War II could point us to events in Germany with the Berlin Blockade and Berlin Airlift. But the USA and USSR didn't really come close to war with each other at that point. The closest the US and USSR came to war against each other came in 1962 with events in Cuba.
The Cuban Missile Crisis was a tense face-off between America, led by President John F. Kennedy and the Soviet Union, led by Nikita Khrushchev, in October of 1962, over the placement of Soviet missiles on the island of Cuba, close to the United States home territory. It is often stated that this moment was the closest the USA and the USSR ever came to the eruption of an actual nuclear war between the two superpowers.
In an address to the nation in October 22, 1962, President Kennedy told of surveillance that revealed Soviet missile sites were observed as part of military build-up in Cuba. He said he would demand that the USSR remove all offensive missiles from Cuba or the US would place Cuba under a "quarantine." A more usual term normally would be a "blockade," but Kennedy avoided that loaded term because that could be interpreted by the USSR as an act of war.
Ultimately, the US and the USSR each backed down in regard to missiles they had in place -- the USSR withdrew the missiles from Cuba, and the USA rather quietly and "voluntarily" removed missiles that the US had placed in Turkey.
