QT is a GUI software development toolkit used by MythTV as a framework for development.
Prior to version 0.22, MythTV was built using QT3. 0.22 will require QT version 4.3 as a minimum.
Most Linux distributions include a package manager, like yum which can install these libraries, and for those that compile from source, the development libraries. A typical command might be
yum install qt4 qt4-devel
Please refer to your OS specific instructions for installing packages or you can get it direct from the source