Comment by selkiesproject

I can only answer for NVIDIA, as that was the platform we developed on, but it should be possible to encapsulate the X11 server within the container without starting an X server on the host machine, and then plug in to the monitor you designate.

I can answer further if you are satisfied with the directions found within https://github.com/selkies-project/docker-nvidia-glx-desktop. It has a lot of effort to NOT require a monitor, but it should also work with one.