TLDR; - See below
And I thought that like Motorola or MOS, Zilog would have a peripheral chip dedicated to Video Generation (like the VIC/VIC-II or Motorola 6847). But I couldn't find anything about such a Chip or Chipset.
Zilog never did any graphics chips, while MOS did a 6845 compatible CRTC - well, plus all the Commodore stuff.
The only thing i was able to find is that aparently most Z80 computers just connected to a TV via RF, but trying RF Modulation on my own without any knowledge of it, would probably not be a good idea.
RF-Modulator is something independent of the graphics chip and related to display connection. Also not really rare. Ready-to-use units can be found all around the net (Example). You'll still need a video source.
So what kind of Video chips would be compatible with the Z80 that wouldn't cause any major timing Issues?
The question is less 'what's compatible' but rather what's available. It sure is possible to use even a C64's VIC2 with a Z80 system, but acquiring one means taking a C64 apart (*1). Also, I think it's safe to assume you're not firm with designing of such systems, so the selection should be easy to implement and handle. There isn't much left in that fits these criteria.
I'd say the prime choice would be a V9958, as it is still available in good quantities at reasonable prices. Even more, there's a good documentation and plethora of Z80 coding examples available, as it was the controller used in MSX2+ (*2) computers.
The V9958 is based on TI's iconic 9918 and fully compatible. So while that one is also still in supply, why go for the lesser? In addition to higher resolutions and more colors the V9958 even includes hardware accelerated commands for line draw and block operations. Even more important for your intended use is a plain RGB output, enabling an easy setup for VGA screens.
Finally there's the V9990. While mostly compatible with TMS9918 and V9958, it offers again improved resolutions (up to 640x480@4 colors), more speed and >120 sprites on screen - a dream for classic tile based game programming. On the negative side it features a QFP package, so you have to like SMD (*3). Considering that most actual chips are SMD anyway not so much of a bad thing. The V9990 is also in good supply, so no need to crack open various game cartridges (*4) .
All TMS9918 family chips offer a great advantage simple (8 bit systems): Video RAM is separated and only a few ports are needed for communication. Address space is a premium on 8 bit systems and not needing some (usually a lot) for video buffers is heaven sent - or the reason why TI did id (beside simplifying access and DRAM control :)).
In a nutshell:
Take a TMS9918 if you want to have the real good old feeling of many limitations and a wired video interface.
The V9958 for a great and simple solution with many features to play with
Or the V9990 if you want to get something that may more fit today's needs - the maximum possible except building your own chip and/or slaughtering an Amiga.
For further reading you might want to take a look at these questions:
Or take a look at the Steckschwein project. Their new video card (*5) uses a V9958. Also a nice example that more generic chips, like video, are less CPU dependent, as the Steckschwein is a 6502 system, where the 9958 was designed for Z80 systems.
*1 - I wouldn't mind getting rid of one more of them - but even after selling the SID on ePay, there's still about the same amount of electronic waste left as before taking apart </DuckAndCover>
*2 - The 9938 used in MSX2 got basically the same features, but might be harder to come by without taking an MSX2 apart.
*3 - The V9958 also got a little quirk to take into account when doing a board, as it's a 2mm DIP (condensed DIP).
*4 - Though, getting a Sunrise GFX9000 or a Tecnobytes Powergraph might be a good way to acquire an 8 bit compatible graphics card, as it's interface is based on the Z80 bus.
*5 - Their first card was TMS9918 based.