add_filter('navigation_markup_template', 'my_navigation_template', 10, 2 );
function my_navigation_template( $template, $class ){
  return '<div class="pagination" id="my-form-pagination"><ul>%3$s</ul></div>';
}

function cars_pagination(){
   $args = array(
        'mid_size' => 1,
       'prev_next' => true
   );

  $nav = get_the_posts_pagination( $args );

  echo $nav;
  ?>
   <div id="form-container" style="display: none"></div>
   <script>
    jQuery( document ).ready( function() {
       jQuery('#my-form-pagination ul li a').click(function(event) {
          event.preventDefault();
          var href = jQuery(this).attr( "href" );
          var obj = <?=json_encode($_POST)?>;
          jQuery('#form-container').append( jQuery('<form />', { action: href, method: 'POST' }));

          for (var prop in obj) {
             jQuery('#form-container form').append(
                jQuery('<input />', { name: prop , type: 'hidden', value: obj[prop] })
             );
         }

        jQuery('#form-container form').submit();
    });
   });
   </script>
<?php

}