The short answer is "It didn't happen."

Missiles are not lone items. There's very little point in just one missile being somewhere because they do require infrastructure to support them. Even the Titan missiles were clustered, although not in the numbers which later systems were put together.

Silos require launch control centers and other things. They're also not protection; they're targets. One doesn't install a silo next to a place to protect. Ever. You don't pull a missile out and say "now THAT's a missile!" to scare the other guys off. Missiles were always kept in very remote areas for an attempt at public safety as much as some attempt at secrecy. Putting missiles in big cites limits the targets. Now you don't have to worry about dividing the warhead strikes between true targets such as population centers and conventional military forces AND silos.

The idea might have arisen purely from the typical sources of legend surrounding someone such as a President, but there might be something else to it. There might have been an air defense missile battery installed, and its purpose was confused by the public. I don't know where the Nike systems were installed, but they WERE placed near population centers as they were used as antiaircraft weapons.