If the target of a feint loses the Quick Contest, is he considered to have used an active defense against it?
No. A successful feint does something to lower or draw the defender’s guard, but the defender does not necessarily execute an active defense in response. Not all feints are fakes. One can fake an attack, but one can also use rhythm (“left-right, left-right, left-left”), stance (a kick after shifting one’s weight as if to punch), deception (look left, strike right) and aggression (beat the foe’s weapon aside) to reduce a foe’s chance of defending. GURPS abstracts all of the above into a “feint roll,” so it would not be altogether realistic to have a feint “use up” a defense.