John Dunning, the author of On the Air, and Jason Loviglio author of Radio Reader both claim that Radio Quiz Shows grew out of the man-on-the street interviews done for the program Vox Pop, carried on 740 KTRH-AM in Houston beginning in 1932. The premise was simple, the random passerby would be asked a question, if they answered correctly they won a prize. The popularity of the program eventually led to network syndication. More here.
The first "true" quiz show was probably Professor Quiz. It debuted on CBS in 1936. The set up was pretty basic. The Guests would ask the host, Professor Quiz, questions. If they stumped him, they won $25. Craig Earl played Professor Quiz. More here.
These shows have an interesting demarcation line in radio history. There was a set of scandals in the late 1950s involving several popular quiz shows on Radio and TV. The most central was the show Twenty-One. Correct answers had been repeatedly given to Charles Van Doren to prolong his 21-week winning streak. Eventually he testified before congress. Here's a sample of that testimony:
"... I asked (co-producer Albert Freedman) to let me go on (Twenty-One) honestly, without receiving help. He said that was impossible. He told me that I would not have a chance to defeat Stempel because he was too knowledgeable. He also told me that the show was merely entertainment and that giving help to quiz contests was a common practice and merely a part of show business."
Prior to the scandals there was no differentiation between quiz shows and game shows. the terminology was interchangeable. Afterwards the radio programs became re-branded as game shows. This was an attempt to distance themselves from the fall out. In general the change was also paired with a general dumbing-down of the programs.
Quiz-like programs continue to this day. They've found a more popular forum on television but NPR programs like "Wait, Wait Don't Tell Me" continue the old tradition. NOTE: There was quiz show parody on KFEL-AM in Denver back in 1951 right before the scandals broke. the audio Oddities Blog posted some archival audio. Check it out.