The problem is you're describing this as blurring so I'm not actually sure what you're seeing at all, because there is no kind of blurring post process that occurs if you have motion blur off. It's worth noting that Quake isn't actually running at 120hz, we can't do that without the physics breaking, but we do interpolate the game world and camera. Maybe you're seeing something that is updating at 60hz without any interpolation? But I'm not really sure what that could be or why you're seeing it as some apparent "blurring". There's only so much I can do with that description, unfortunately.
A video showing off this blur, if at all possible, might be the key. Maybe use a high speed camera (they tend to be on phones now)?
Also, do you get this if you play with a mouse and keyboard instead?