How to run C++ applications on a webpage?

Check out NaCl (Native Client). No Qt, but you can use c++.

You can't. You can write the server side in a web app in C++, of course, but the Qt GUI framework is only for applications that show their user interface on the machine they are running on. (With the usual caveats in the case of remote X11 connections and so forth, but that won't help you for a web app).