The cold war was not a conflict where sides shot at each other. Instead, they called each other names. However, it was a weapons race. Each side was trying to develop better weapons so that it could win if there ever was a real hot war.
No, the Cold War is not typically considered to have been a military conflict. There were, of course, many military conflicts that did arise during the long period of the Cold War. However, the Cold War as a whole is not seen as a military conflict.
The Cold War certainly included some military conflicts. The most famous of these were the Korean War and the Vietnam War. There were also other conflicts that were connected to the Cold War such as the Russian invasion of Afghanistan and many civil wars like the one in Angola. However, these were only parts of the Cold War.
The Cold War as a whole was a multifaceted conflict. It involved military conflict, but it also involved political and economic conflict. It involved competition in athletics and competition for who could get into space first, or to the moon first. We would be wrong to say there was no military conflict, but we would also be wrong to say that the Cold War as a whole was a military conflict.