If you have access to a Windows machine, it might be simpler to install mingw on it and compile on it directly. (And mingw's gcc won't generate binaries that depend on cygwin1.dll.) ...

If you still think that operating systems are tied to hardware, you need to update your thinking; nowadays there are lots of options that let you run one operating system within another or in parallel ...

You would need to purchase a Windows license from TrollTech because the Qt/Windows Free Edition only supports MSVC++. The KDE on Cygwin project has gotten Qt/X11 Free ...