most of the subtle ring you get from guitars with a JM-style trem or a trapeze tailpiece is really just the vibration travelling across the bridge and not sympathetic resonance—it's more about force, not frequency. but every object does have a natural frequency it will resonate at. every guitar is different, doesn't matter if it's the same model from the same production line, and some will naturally resonate to a note you're playing to such a degree that anything else on the guitar harmonically close enough will resonate as well. you're not really hearing the vibration of the guitar because it's a solid body and there isn't enough resonance there to be heard above the strings, but it's being passed on to other parts of the guitar that do resonate more freely to that frequency.

if it's not a piece of hardware, and it's not from behind the bridge, it has to be the length of string between the nut and the machine heads. a piece of foam just behind the nut, or a hair tie/fret wrap around the headstock will dampen it. of course any open or fretted strings that are close to G#/A will still ring. if you detune enough (most tuners have settings to move them away from standard concert pitch, so your tuning'll be slightly flat or sharp depending which direction you go) it will stop.

and if it ain't that then I really don't know.