How to convert AV to HDMI

HDMI is just "DVI with knobs on" on the video side.

As to "can it be done"... my first question is "what FPGA"? Some of them can create HDMI/DVI signals with the IO blocks, others just fundamentally can't.

DVI uses TMDS signalling, which is an encoding on top of a Current Mode Logic (CML) differential pair. CML is actively pulled down by a current source for a '0' and floats high with a termination resistor at the far end for a '1'. It might be emulatable for a hobby project by using a bidirectional LVDS pair driven low and using the tristate line to drive and release (a bit like doing an open-drain drive).

Then you have to encode and serialise the data. TMDS describes how to encode the data bits, and then you you "just" have to serialise the data bits across the data pairs. The specification can be found here - see section 3:

Digital Visual Interface Spec

The TFP410 chip data sheet also has a reasonable description of what goes on:

TFP410 - TI PanelBus™ DIGITAL TRANSMITTER

Get a BlackMagic PCI Express HDMI video capture and pass-through card. As long as the source is not HDCP encrypted, this would be pretty simple, assuming you're OK with Windows as the host (I don't know of Linux drivers.) There are other video capture/edit suites that can do it as well, but generally more expensive (Matrox has some low-end ones for example.)