It really is not correct to say that there were only two countries “involved in the Cold War conflict.” Many more countries than that were involved significantly and practically every country in the world was involved in at least some peripheral way. However, if you have to say that there were two countries involved in this conflict, the two that were most important were the United States and the Soviet Union.
The Cold War was a conflict between communism on the one hand and capitalism and/or democracy on the other. Almost all countries in the world had to take sides to some degree, which meant that everyone was involved in some way. For example, in the 1980s and early 1990s there were civil wars involving communists and non-communists in such diverse places as Afghanistan, Nicaragua, and Angola. These civil wars were “proxy wars” in the larger Cold War conflict. What this shows is that even relatively small and obscure countries like these were part of the Cold War.
However, even though essentially all countries in the world were involved in the Cold War to some degree, the US and the Soviet Union were the most important countries. The United States led the non-communist side of the conflict, trying to prevent communism from spreading. The Soviet Union led the communist bloc, trying to spread communism throughout the world. If you have to specify only two countries that were involved in the Cold War, these are the two to name.