Can you check something for me please? I'd like to try doing a "Run as administrator" on your iTunes.
Quit iTunes first.
In Computer, open Local Disk C: or whichever drive you have your program files installed on.
If you have a 32-bit Windows, open the "Program files" folder.
If you have a 64-bit Windows, open the "Program Files (x86)" folder.
Open the "iTunes" folder.
Right-click on the iTunes.exe file and select "run as administrator".
Try playing the movie trailer in the Store now. Does it play for you without the message?
The problem is not within iTunes - the trailers work in that program. The problem I am having is with the iTunes Movie Trailer website. Quicktime works fine by itself and in the iTunes program, but when I try to watch a trailer from the website it's like I don't even have Quicktime installed on the computer. At that point a message pops up asking me to download Quicktime.