Hi, fist of all, it would be nice, if you could tell us your hardware. Some notebooks are using two graphic cards One in the CPU and one external. Mine for example. has a NVidia card as external and an Intel chip built in the CPU.
Those are switched in linux with "bumblebee". If we correctly know the hardware, we can explore more. The HDMI output is sometimes controlled by the hardware (on my desktop computer it is the driver from the external graphic card, which in my case is also an Nvidia card). I suppose, Intel chips and AMD chips are doing the same. However, some intel GPU drivers are built in the kernel, others need external packages. So, let us know more and see, if we can help. Best regards Hans