How to get change event working of a taxonomy select box on a node form?
This isn't really a Drupal question, but at a guess I'd say the problem is that you're binding the event to an element before the DOM has loaded...so the element doesn't exist yet. You should use one of the standard jQuery methods to run your code when the document is ready, e.g.:

$(document).ready(function() {
  $('#taxonomy-select-box-id').change(function(){
    alert('change');
  });
});