By default any calls done by Qt over HTTPS will fail. This is because the OpenSSL it relies on is not included into the distribution that was installed. All you have to do, to make it work, is to download OpenSSL 1.0.xxx library for Windows from here and install to any folder (for example: “C:\OpenSSL”). The tiny 3MB package is enough. Then copy the libeay32.dll and libssl32.dll into developed applications folder (the output one) and it will be automatically picked up during startup.
There is no need to mess anything with Qt at all (and for sure no need to rebuild Qt framework!).
It was also described on StackOverflow, too.