Checked Mersey Tunnel regs, Solo Motorcycles are free, with sidecar or three wheelers have to pay, I don't think they've built quads into it as the next listing is for PLG, which it's not.
If nothing else it'd make for an interesting argument with a mole.
Pretty sure most quads are registered as PLG, occasionally you see the odd one as agricultural vehicle (which means very chap tax and no mot ever) but this is rare that i've seen, and not sure how they manage it.