You may do this:

1. insert a <form> tag with an id, insert all your hidden <input> (optional) and insert an id to the <button>

<div class="modal-body">
  <form id="contact-form" name="contact-form" action="#" method="POST">
    <input type="hidded" name="myvariable" value="myvalue"> (optional)
    ... your modal content
    <button id="contact-button" class= ... >
  </form>
</div>

2. insert this javascript on document ready or online inside a <script> tag

$("#contact-button").on('click', function() {
  $("#contact-form").submit();
});