No, there was no detonation of missiles during the Cuban Missile Crisis, which occurred in October 1962. The crisis was a 13-day confrontation between the United States and the Soviet Union over the presence of Soviet ballistic missiles in Cuba. It brought both superpowers to the brink of nuclear war.
Ultimately, diplomatic negotiations led to a resolution of the crisis. The U.S. agreed not to invade Cuba and secretly agreed to remove its missiles from Turkey, while the Soviet Union withdrew its missiles from Cuba. The tensions of the crisis highlighted the dangers of nuclear confrontation, leading to improved communication between the two nations, including the establishment of the Washington-Moscow hotline.