Google's add-on store has converters that read the open document and hand back Markdown. The best-known is Docs to Markdown; install it from Extensions → Add-ons → Get add-ons, then run it from the Extensions menu.
- Works on the document as it stands, no download first.
- Usually keeps headings, lists, links, bold and italic.
- Needs an add-on with access to your document, which is a permission decision worth making deliberately.
- Output quality varies with tables, comments and nested lists.