Plywood will not stain in the same manner as the QSWO because the veneer is so thin and there is glue immediately behind it preventing even absorption. You probably can get QSO plywood, from a specialty supplier, but why not just glue up the panels instead.
Sounds like you will be going to a hardwood mill regardless, so just have them resaw some into 1/4" or 3/8" thick boards.
I should have done that for this Mission Style headboard, but I didn't know the issues involved. You can see where the stain and grain pattern is not really compatible with the Quartersawn if you look closely:
http://www.woodworkingtalk.com/f2/mission-headboard-build-37911/
I would limit the use of plywood to drawer bottoms and backs and not use it on the sides. :no: