Using Handbrake the answer would be no. Handbrake is a CPU-only h.264 encoder.
The software you are actually looking for is Badaboom which can use Nvidia and Intel graphics processors to speed up the encoding process. Most GPU encoders seem to use CUDA, which is almost exclusively an Nvidia technology. The question linked by slhck mentions MediaCoder, but it requires either an Nvidia graphics card or an Intel CPU with a 2nd generation graphics core. MediaCoder is also a bit "complicated" and suffers from the curse of a thousand features...
ATi supply their own video encoder that used to be called "Avivo" but is now the AMD Codec Package which includes the AMD Video Converter and you may have some luck using it, I have no ATi/AMD card to try it with.