There is a Stirling engine design which only has a piston as the single moving part, called a "lamina flow Stirling engine".

Note that while implementations usually show a crank and rod with a flywheel, it could just as easily use a magnet and coil to generate electricity.

That get's you down to a single part.

Then you have this:

https://en.wikipedia.org/wiki/Thermoacoustic_heat_engine

...that gets you down to something that can generate sound from a heat differential, and you could couple that sound to some kind of transducer to generate electricity. Still a moving part, though.

You probably can't get zero moving parts and yet have it do useful work, but you can get really close I think.