Clean URLs don't work after install in Backdrop
If Clean URLs are working in Backdrop, but not in Civi under Backdrop, then the problem is almost certainly in civicrm.settings.php.  My Backdrop/Civi sites (and also Drupal 7/Civi sites) have this snippet of code in civicrm.settings.php:
if (!defined('CIVICRM_CLEANURL')) {
  if ( function_exists('variable_get') &amp;&amp; variable_get('clean_url', '0') != '0') {
    define('CIVICRM_CLEANURL', 1 );
  }
  else {
    define('CIVICRM_CLEANURL', 0);
  }
}

I suspect you're either missing this code, or perhaps there's something wrong with your &quot;clean_url&quot; variable in Backdrop.
Try replacing that with just this and let us know if it solves the problem?
define('CIVICRM_CLEANURL', 1 );