From my exposure to people deploying QtWebKit or WebKit/GTK+ there are some things that re-appear and I would like to discuss these here.
The best way to find out if you suffer from this problem is to use -E instead of -c to only pre-process the code and see where the various includes are coming from. It is a strategy that is known to work very well.
Most likely you do not have a DNS Server set, or no networking, or the system your board is connected to is not forwarding the data. Make sure you can ping a website that is supposed to work, e.g. ping www.yahoo.com, the next thing would be to use nc to execute a simple HTTP 1.1 get on the site and see if it is working. In most cases you simply lack networking connectivity.
It might be either an issue with Qt or an issue with your system time. SSL Certificates at least have two dates (Expiration and Creation) and if your system time is after the Expiration or before the Creation you will have issues. The easiest thing is to add ntpd to your root filesystem to make sure to have the right time.
The possible issue with Qt is a bit more complex. You can build Qt without OpenSSL support, you can make it link to OpenSSL or you can make it to dlopen OpenSSL at runtime. If SSL does not work it is most likely that you have either build it without SSL support, or with runtime support but have failed to install the OpenSSL library.
Depending on your skills it might be best to go back to ./configure and make Qt link to OpenSSL to avoid the runtime issue. strings is a very good tool to find out if your libQtNetwork.so contains SSL support, together with using objdump -x and search for _NEEDED you will find out which config you have.
This is a pretty common issue for WebKit/GTK+. In WebKit/GTK+ we are using GIO for local files and to determine the filetype it is using the freedesktop.org shared-mime-info. Make sure you have that installed.
This is another issue that comes back from time to time. It only appears on WebKit/GTK+ with the DirectFB backend but sadly people never report back if and how they have solved it. You could make a difference and contribute back to the WebKit project.
In general most of these issues can be avoided by using a pre-packaged Embedded Linux Distribution like Ångström (or even Debian). The biggest benefit of that approach is that someone else made sure that when you install WebKit, all dependencies will be installed as well and it will just work for your ARM/MIPS/PPC system. It will save you a lot of time.