This might be of help.
http://www.analog.com/library/analogDialogue/archives/47-02/HDMI_VGA.htmlIt uses an AD9983A as an 8 bit display interface feeding that to the HDMI-TX (ADV7513)
http://www.analog.com/en/products/audio-video/analoghdmidvi-interfaces/ad9983a.htmlBut this needs the RGB analog input plush the HSync/VSync, so you might need a 3rd chip an ADV7125 330MHz Triple 8-Bit High Speed Video DAC.
http://www.analog.com/en/audiovideo-products/video-encoders/adv7125/products/product.htmlBut making the board might cost you more than using something that already outputs HDMI, using that Raspberry Pi zero ($5 if you can get your hands on one that is) and program it to be used as a plain frame buffer with probably even audio might be a better way.
Or the FPGA route you can find a board with HDMI out like this one that uses the ADV7513 HDMI-TX chip.
http://www.terasic.com.tw/cgi-bin/page/archive.pl?Language=English&CategoryNo=165&No=830&PartNo=1But with the caveat that it doesn't have the audio hooked in, also I believe the samples use the NIOS-II softcore and I don't think they have a bare metal sample driving the ADV7513.
I did write a review a while back about that dev kit, but it's not cheap at $179 + shipping but for that it has, not expensive at all.
https://www.eevblog.com/forum/microcontrollers/terasic-cyclone-v-gx-starter-kit-(c5g)-review/msg402667/#msg402667But if you stick to VGA output then there are more choices like a PSoC5LP devkit driving a simple R2R VGA output module.
Links to the boards in this post: (Edit: fixed link to right one)
https://www.eevblog.com/forum/projects/make-use-of-an-old-cga-monitor/msg696341/#msg696341Them working here:
https://www.eevblog.com/forum/projects/make-use-of-an-old-cga-monitor/msg697201/#msg697201But you will be limited to a small framebuffer unless you add some external memory.
You can also get a DE0-Nano and this board:
http://www.wayengineer.com/digiasic-fpga-adv7125adv7123-vga-board-p-160.htmlAnd adding some kind of frame buffer and some communication to your z80 to address that frame buffer.
But I guess it all depends what video content you have in mind, if it's just a terminal with just text that doesn't require many colors, the PSoC5LP with the R2R DAC VGA module might be the cheapest and easier to do, but by easy you need to learn how to program the PSoC digital components, then you can add some ram to it and then use the VGA output to convert it to HDMI.
The simplest would be an lcd with frame buffer and a controller built in.
And again, none of the above deal with the HDMI audio as far as I know.
Edit: fixed link to sources for PSoC5LP and R2R DAC VGA module above (had a duplicate link to the fpga dev kit)