Well, I’m embarrassed. After reading the ACF source code I’ve found a display_errors
function.
To display the errors messages you just need to add it into the callback of the POST request, like this :
$.post(
my_ajax_obj.ajax_url, data,
function(result) {
errors = result.data.errors
if (errors === 0) {
// Code for success
} else {
acf.validation.display_errors( errors, $("form.acf-form") );
}
}
);