Hey,
Polylang doesn't have a way to do this. It can't do array string translations. So to get this to work you need the premium theme where you can use shortcodes to create the icon menu and place those shortcodes in the page content where polylang can do translations. The free theme doesn't have this shortcode option. So it your sticking to premium theme you would have to code the icons into the page content yourself.
Sorry this isn't more helpful but polylang doesn't have a quick tag option like qtranslate.
Kadence Themes