USB to HDMI for PS4, will it work?

I STRONGLY doubt it. It's not impossible, but I would think if it were to work, it would have to be an adapter/converter specifically for the PS4

USB is not inherently a port used for video transmission. Any device that connects is going to need a driver so the system knows how to use it. And that would require that a driver exists for whatever Operating System the host device is running. Which would mean the adapter would need a driver specifically for the PS4 Operating system. Typically these adapters will list OS requirements in the product description or on the box/manual and I don't know that I have ever seen this particular sort of adapter before for a console.

If your issue is that the HDMI port is not functioning, your better bet would be to find someone who can repair that. Lots of folks who do console repairs, and HDMI port repairs are very common on any modern console.