I am developing cross-platform Qt application. It is freeware though not open-source. Therefore I want to distribute it as a compiled binary.
On windows there is no
You can also distribute Qt shared libraries on Linux. Then, get your software to load those instead of the system default ones. Shared libraries can be over-ridden using the LD_LIBRARY_PATH environment variable. This is probably the simplest solution for you. You can always change this in a wrapper script for your executable.
Alternatively, just specify the minimum library version that your users need to have installed on the system.