Is there a way to change the font family on Markdown? Specifically, for the
code and regular text. I've seen this feature in Markdown editors and viewers, but I'm not sure if it can be exported.
As far as I know, nope.
Practically - SE supports whatever CommonMark does and fonts are system fonts. While there's a subset of supported raw HTML - I don't think it supports, nor will it support the style tag, unless the developers have been up to things secretly. You could possibly change things client side, but there's no way to set specific font families 'globally'.