This is debatably one of the funniest moments in the history of humanity. This clip alone has more complexity than any other form of literature and art in the modern era. This is extremely high IQ so if you don't find this funny, you're giga- low IQ.
Quagmire already has an exaggerated jawline that protrudes in every direction, even having a butt chin (keep this in mind). Shortly after Quagmire appears on screen, his mouth begins to morph into a basin, where his body proceeds to transform into a toilet right after. His transformation has lead him to become a toilet, with the basin being his mouth and still having his human eyes and nose. Quagmire toilet then proceeds to flap it's basin lid up and down, this is simulating the way humans talk by opening our lips. The comedic layer of this comes from the essence of randomness, we did not expect Quagmire to transform into a toilet, thus the comedy is when he unexpectedly turns into a toilet.
There is also high IQ foreshadowing with Quagmire's butt chin, butts are commonly associated with toilets. This was a slight warning by the writers that he was going to turn into a toilet, perhaps planned from the early character design concept.
@mental_out @Giegue @Animecel2D thoughts?
