Generate documentation as markdown files using doxygen
I know this question is old but there seems to be a new tool available: https://www.npmjs.com/package/doxygen2md
I don't know of anything that will do it directly, but doxygen has the ability to output an XML document using the XML generator.
There are various projects out there that can then consume said XML data to transform it, one of the simplest ones that I am aware of is Breathe which is used to convert the XML documents to something that Sphinx can use to output documentation using doxygen as a pre-processor so to say.
The code for Breathe may be simple enough to use as a starting point to then generate Markdown files instead of the ReStructuredText files that Sphinx requires. I haven't looked at the code for Breathe, but I have used it for some major projects.