Using Handbrake the answer would be no. Handbrake is a CPU-only h.264 encoder, though it now is able to use OpenCL for video scaling and has some support for DXVA for video decoding.
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.
There are a couple of older products that have been obsoleted, specifically Badaboom and the ATi Avivo software.
Badaboom which could use Nvidia and Intel graphics processors to speed up the encoding process but it is now end-of-life and does not support newer graphics cards.
ATi had their own video encoder that used to be called "Avivo" before being merged into the AMD Codec Package which included the AMD Video Converter though it is no longer available.