In Star Trek: Voyager's "Death Wish" Q has blue lips. Is this in all appearances in Star Trek (TNG, Voyager,...) the case? Also, why are the lips blue?
A quick search of Q images reveals that he doesn't always have blue lips. It appears he tends to change his complexion depending on the role he is assuming.
That said, the picture below with the blue lips might be considered Q in his natural state, so perhaps they are indeed blue. As to why, I have no idea other than the fact that he is an omnipotent alien, and perhaps that is one of their features :)
I never noticed his lip color: The specifics of skin, hair, and eye color are especially superficial in the Star Trek universe.
I expect any unusual coloration is to reinforce a being's other-worldliness. Q is definitely not human and should not be confused as such. He may as well have a diagonal purple mohawk.
He doesn't eat, once called it a 'vile human habit'. So it can be assumed he doesn't have a circulatory system or his circulatory system is simply not working which would normally feed his body nutrients.
With this, he can also survive in space unassisted (numerous episodes) - which suggests he doesn't breathe or process oxygen.
Blood needs oxygen replenishment to turn from blue to red. As well as a functional circulatory system that replenishes the nutrients which further discolors the blood to blue. Neither of which he has.
So his lips. Without 'red blood' circulating, are blue.
He's immortal. I suspect that's just another way of saying he's undead.
A pretty fantastic attention to detail on the writer's parts, don't you think?