Make sure software is up to date. Updated versions may implement support for modern file formats. This can cause QCT files to not be compatible with QuarkXPress. Previous versions handled file formats just fine. These should also open in QuarkXPress.
The QCT file format is associated with Memory-Map. QCT files can be saved and opened in Memory-Map. Psyberia AlpineQuest can also open them. Memory-Map works on Windows, macOS, Android, and iOS. Windows QCT maps are typically in C:Maps or C:Maps_<version>.
Other programs that open QCT files are available. The QCT file is for high resolution map images. Its key goal is low processing overhead and compression.
The qct extension relates to QuarkXpress page layouts. These template files apply in QuarkXpress. Work with them there.
QCT files should only be edited in appropriate software like Memory-Map. Associate QCT files with the right application. Update software that should open them.
Memory-Map