No.
The U.S. military considered but ultimately decided against staging offensive ballistic missiles in Alaska due to the cost of building such facilities in an area with poor logistics. In addition, improvements in missile technology meant missile launching positions could be located in the Lower 48, where distance lowered the chance of a "Pearl Harbor-style" pre-emptive strike.
Nuclear weapons were deployed to Alaska as cruise missiles to be launched from aircraft, as air-to-air missiles, as torpedoes and other naval weaponry, and as surface-to-air weapons at the various NIKE air defense sites across the state.
The Soviet experience was similar, though it actually did build an offensive missile base in the mid-to-late 1960s in the Chukchi region. It was located near the airbase at Anadyr, but as with the Americans, the Soviets found operations in the far north too expensive and maintenance-intensive. The Anadyr missile base was shut down before the end of the 1960s.
On the Diomedes, the only military presence were border stations ─ even listening antennae tended to be mounted on the mainland, as maintenance on the Diomedes was a huge hassle.