Your best bet would be to try installing the compiled version of Qt for your linux flavor. If you have a working internet connection you can try installing it using "Synaptic" on Ubuntu and using "Yum" on Fedora. Synaptic and Yum automatically download and install the dependencies so you don't have to worry about 3rd party dependencies and compilation etc...

If you still want to install it from source, you have to first install all dependencies. Best way is to keep on trying and installing dependent packages as errors appear during compilation.