Different font for and only

This depends on your theme, and where and how these tags are used on your site.

If you can provide the URL of a page on your site with these tags in use, I’m sure I can give you custom CSS to do this. Also, what font do you want to use?

George is right, it depends on your theme, but generally speaking you could add this to the Additional CSS customizer section:

h5, h6 {
  font-family: "Times New Roman", Times, serif;
}

Use whatever font list you want. If it’s not a common system font, you’ll need additional code to load a web font. The above CSS will affect every h5 and h6 header in your site, unless there is more specific CSS that overrides it.

The !important modifier can override some overrides. There’s no way to override element style attributes other than to alter the attribute itself.

If you want to alter the font of headers in a specific context, sharing an URL like George suggested is the best path forward.