Definitely not my area of expertise, but coax (to the best of my knowledge) is analog only. I don't believe you can convert it to digital.
Your cable box takes the analog signal, interprets what you have converts to a digital output.
I'm making an assumption here, but your believe is if you can connect the coax to the TV via HDMI you can view/control the channels you receive? It doesn't work like that. The cable box is a key component as far as what channels you can receive/view.