The Bury boxer remains undefeated with the stoppage, dominating both rounds in the Manchester Arena.
Quigg knocked down Martinez twice in the second round, the first time the Englishman caught the Spaniard with an uppercut, dazing Martinez and then unleashed a flurry of hooks, knocking him down in the centre of the ring.
That spelled the end for the challenger, as he got back on his feet to come back and try to regain his composure.
Martinez's defense didn't last long after that, as Quigg continued to ruthlessly attack with continuous left and rights, before landing a right hook on to the jaw of the 28-year-old.
Quigg remains undefeated after 33 fights, winning 31 of them and drawing twice.