This is a matter related to ‘aadab’ and can vary from person to person depending upon the situation as well as their love and respect for the message of the Quran. The key thing is to keep it accessible enough to where you can pick it up often and read it anytime you want. One can be very respectful and place it in a secure high place, however if it keeps him from the purpose that the Quran was sent for (to be read and understood) then that respect is only for the ink and the paper. When someone truly appreciates the wisdom contained in the Quran, their respect undeniably shows in their deeds, one of which could certainly be placing it in it’s own dedicated space.