The coax is no good for hdmi. I know of no commercially available products that will convert HDMI to some of signal that can be transported over coax.
Obviously, someone can build and engineer as system to encode hdmi signals to some of form of RF quadratures that can then be transmitted over coax just like docis 2/3 cable modems, but none of that stuff is readily available for home endusers.
So TLDR... nope it can't be done feasibly or cost effectively.