Not that this is exactly textobj-quote's fault, but if easyclip is configured as follows (which is what the easyclip documentation reccomends):
then ReplaceWithCurly and friends don't make any change to the text because pasting no longer uses the " register. It seems that using set clipboard=unnamedplus fixes this problem, although I haven't fully worked out if it has other knock-on effects.
I guess textobj-quote could check for easyclip's existence and alter the paste register it uses; or maybe even documenting the issue might be enough?