No replies yet so I'll try to offer something.

There is a lot of uncertainty when exploring what-if scenarios. You can't be sure what may have happened without the efforts of the spies involved. The best example I can give of spies potentially contributing to the peaceful resolution of the Cold War might be the Soviet spies in the Manhattan Project. This may seem counter intuitive, but I'll try to make the case.

Combined, Klaus Fuchs, Theodore Hall, and David Greenglass provided Soviet intelligence with a wealth of valuable information from the Manhattan Project, which likely sped up the Soviet development of nuclear weapons significantly (although by how long is hard to say).

The foundations of the cold war were in place even before WWII had ended, but it was the period from 1946-1949 that really cemented the standoff and lead American policy makers to give up on cooperative engagement. Also during this time the US had a monopoly on nuclear weapons. Despite this, the US did not yet have a large enough nuclear arsenal to be confident that it would be decisive in a war with the Soviet Union. It would not be long though before the US had amassed a huge arsenal. However, when the Soviet Union successfully tested its first atomic bomb in 1949, the US nuclear monopoly ended, as did any real risk of the US using its nuclear weapons to push the Soviets back.

Although the two great powers came very close to the brink of Armageddon a few times, mutually assured destruction effectively made nukes useless in practice, and the stability of mutually assured destruction lasted throughout the remainder of the Cold War. The success of the Soviet spy efforts around the Manhattan Project brought that stability about faster, and there is no telling what could have happened in an alternative history where the US nuclear monopoly lasted another year or two.

Of course, while the Soviet Union's shiny new nukes kept both powers at arm's length, by restraining the US nuclear threat, it also opened the door for conventional warfare. It wasn't long after that North Korea, with the blessing of Stalin, invaded the South.