Quadraphonic sound was one of the earliest consumer offerings in multichannel audio, (i.e. more channels than stereo) It was introduced to the American market in the early 1970s. Quad was not a single format; but actually an array of several different incompatible formats. Quadraphonic media was available on vinyl records, eight tracks, and reel-to-reel and yes, even broadcast radio. It was uncommon, but some brave and bullish broadcasters bought into it. Some Quad Hi-Fis can be found here
In Seattle back in 1977, an experimental Quadrophonic broadcast was attempted at 92.5 KZAM and 98.1 KING. In conjunction with Windham Hill records. In a very MacGyver-ish approach, this required two separate stereos and tuners. Each station broadcast two channels of the quadrophic audio. If you did, you got to hear 360 degrees of John Klemmer. [Not sure if that's worth it] (KZAM is now KLSY)
In Chicago 97.9 WLUP broadcast one live quadraphonic concert in 1979. It featured Joe Jackson, Rainbow and the Blue Oyster Cult.
In Detroit Quadrophonic radio was attempted by several stations 102.9 WWWW, 101.9WDET, and WABX at different times.
As early as 1975 WQIV in New York City attempted full time quad broadcasting. They broadcast live concerts, including the earliest syndicated Rock Around the World shows, whole albums, and other things unusual at the time on commercial radio. It was very much a product of the free-form FM era. It eventually flipped to become classical WNCN, the calls and format it had previously. (Currently 104.3 WAXQ )
In 1969, WCRB radio in Boston collaborated with WGBH to present a series of Boston Symphony concerts live in quadraphonic sound. Two channels being transmitted on WGBH and the other two on WCRB. Oddly WCRB continued to broadcast some of its recorded music in quad for several more years.
In 1979 Amex tape did promotion with San Francisco's KRON and KIOI with KRON broadcasting the front two channels and KIOI taking the rear two channels. For this first Quadraphonic broadcast, The Greatful Dead performed. It was televised on KQED and simulcast on KSAN.
Quad was killed by the advent of the CD and completely stalinized with the birth of surround-sound. Skip Pizzi at RW online continually writes propaganda in favor of surround sound for radio. I have not bought into this cult yet. Radio has been struggling to push HD radio for 10 years now, I doubt surround sound will fare any better. Feel free to read up and judge for yourself [Link here]
...And here is the surround sound system that could have been, The Ambsonic Sound System...