I've picked up many a rock from deep caves that have quartz mixed with other material that is less hard. It certainly got that way from being in heat high enough to make it molten - as was the material it was mixed with.
But no process I know of would make quartz in molten state separate from other material and become a predominate layer that would explain earthquakes - simply because quartz is among the least materials found at great depth in any of the great open-pit mines around the world. (I've been to several, as a visiting 'rock hound' - begging for 'finds.')
Quartz has a hardness of 9 - just below that of diamond, but it can be shattered just like glass, with great enough pressure. (So can diamonds) But, before I believe all quakes are caused by the presence of quartz, I'd like to see a great deal more of it in the deep mines around the earth - because there just isn't that much that I've seen.
Tectonic plates subverting under another plate causes movement, simply because at 20 miles down, all material becomes molten - allowing that movement. Sorry, quartz just ain't IT when it comes to earthquakes, IMHO.
Hardness of 9? What scale, on the Mohs scale it’s 6-1/2 to 7...
Like this one:
New measurements of quartz abundance from EarthScope data show that mountains are quartz-rich (red-orange colors). The Great Plains, Columbia Basin and Great Valley have little or no quartz (blue-green colors).
Credit: Lowry & Perez-Gussinye