But the band, whose recent album, "Hail to the Thief," has been praised by critics in Britain and the United States, didn't attend Thursday's ceremony, sending a video message instead.
The film was soundless, but in type at the bottom of the screen it poured scorn on the magazine for branding them "miserable." The short video ended with the line: "Radiohead aren't talking to Q."
Readers of Q had selected Radiohead over Coldplay, Foo Fighters, Red Hot Chili Peppers and Blur.
Coldplay, also no-shows at the ceremony, had received the most nominations, up for best single ("Clocks"), best video ("The Scientist") and best live performers as well as the top prize. But they missed out in every category.
However, Christina Aguilera was on hand to pick up her best single award for "Dirrty."
"It's about time 'Dirrty' got some love. Thank you to Q for recognizing the woman with the balls to make this video and loving every minute of it," the 22-year-old singer said, referring to the raunchy video that accompanied the song.
Other winners included Electric Six, best video for "Gay Bar"; Robbie Williams, best live act; Blur, best album for "Think Tank"; The Thrills, best new act; The Cure, the inspiration award for their influence on music; and Duran Duran, lifetime achievement.