It was first seen slung over Ford's back in "Not What He Seems".
In "The Last Mabelcorn," a prototype of the Quantum Destabilizer can be seen sitting on Ford's desk in his room near the beginning of the episode.
In "Weirdmageddon Part 1," Ford takes the weapon up into the church's clock tower on Main Street near where Bill Cipher and his Henchmaniacs are. As Ford prepares to fire at Bill, a weirdness wave washes over the tower, causing the bell behind Ford to come to life and yell, distracting Ford, who misses and only hits Bill's hat.
It resembles a large gun. It has a long gray body and a bright blue bulb at the end which emerges when the weapon is activated. It also has a scope which is used to aim. The reticle is shaped like a triangle.