I have a website in which some pages have user-generated content. I use phantomjs (based on QtWebKit) to take screenshots of the pages, so they must render perfectly in QtWe
If the problem is that the chosen font doesn't contain the specified characters, I would recommend that you embedd the font into you website. With that you can ensure that every platform is able to render your content.
You say that Chrome doens't have the behaviour you described as incorrectly. So the cause must be in your old WebKit implementation. You may try to update to Qt 5 wich uses WebKit 2.0, like Google Chrome. Qt 4.x uses the old WebKit 1.x versions.
Bugfixing the WebKit code is a very bad idea. The codebase is around 2 million LOC and you need a 64bit toolchain to compile the QtWebKit module. It is also very likely to get compiler errors if you do not follow the instructions exactly. I had my self the need to debug the WebKit code because of a bug in the GIF image renderer. This is the real hell on earth ;)