Quire rhymes with:
inquire, mcquire, reacquire, squire, barbed-wire, dwire, inquire, maguire, mcguire, mcgwire, mcquire, o'dwyer, reacquire, squire, swire, wyre
Quire sounds like:
qasr, quackery, quaker, quarre, quarry, quasar, queer, queerer, querry, query, quicker, quiry