A couple of entries from Song Facts re "Quinn", all just speculation I suppose-

The Grateful Dead occasionally played this at their shows. Here's one story that circulated about the song: The Grateful Dead years ago had a wild LSD party in a New York City hotel during a tour visit. Allegedly, one of the party guests was Bob Dylan. One of the other guests at the hotel didn't appreciate the noise and voiced several complaints. It was actor Anthony Quinn who'd played an Eskimo in The Savage Innocents. That could have inspired a partying Dylan to write a strange and funny song like this. >>

One theory is that "The Mighty Quinn" is Sheriff Larry Quinlan, who raided the Castillia Foundation land in Millbrook, New York and arrested Dr. Timothy Leary and his group of hippies. Quinlan confiscated all the LSD and other drugs at the scene. In this scenario, the "pigeons" are informers. >>