@JeremyECrawfordA quarterstaff doesn’t qualify for two-weapon fighting (RAW & RAI). As a DM, I’d allow it with a good story reason.
That is a very interesting stance considering polearm mastery.

— Ace (@WolfsGoRawr) July 10, 2015

@WolfsGoRawr The Polearm Master feat intentionally doesn't use, or even refer to, two-weapon fighting (PH, 195).

— Jeremy Crawford (@JeremyECrawford) July 10, 2015

@WolfsGoRawrI know. But using the TWF for a quarterstaff seems like a cheesy way of getting that effect from polearm feat!
That's part of why the rules aren't written to allow it.

— Jeremy Crawford (@JeremyECrawford) July 10, 2015