In Chapter Nineteen of Order of the Phoenix we have the following statement from Professor McGonagall:
“I think you’ve got enough to be getting on with at the moment,” she said loftily. Nobody could quite believe their ears until she looked directly at Harry and Ron and said grimly, “I’ve become accustomed to seeing the Quidditch Cup in my study, boys, and I really don’t want to have to hand it over to Professor Snape, so use the extra time to practice, won’t you?”
This would seem to indicate that the Quidditch Cup is stored in the office of the head of the house that won it until the next house wins it at which point it would be transferred to the office of the other head of house.
However, in Chapter Seven of Chamber of Secrets we have the following statement from Ron, describing his detention in the Trophy Room:
"Fourteen times he made me buff up that Quidditch Cup before he was satisfied. And then I had another slug attack all over a Special Award for Services to the School. Took ages to get the slime off... How was it with Lockhart?"
This seems to indicate that the Quidditch Cup is stored in the Trophy Room. Now one could perhaps argue that there is in fact more than one Quidditch Cup; but if that is the case then there would be no need to pass it around between the heads of houses as they could just give the new winner a new cup.