<input type="text" class="form-control" name="moms_md_name" id="moms_md_name" value="">

var mom_maiden_name = $('#moms_md_name').val();
try 1
} else if (maiden_name !== '' && maiden_name !== null) {
      if (!maiden_name.match(/^(?=.*[a-z]).{3,20}$/)) {
         alert("Mother maiden name not properly entered!");
   }

blank/empty - work (get result)
disable jquery validation get result
enter wrong input 222/aa - work get (get error msg)
enter correct input - no result at all <-?

try 2
 }  else if(maiden_name !== '' && maiden_name !== null){
                    if ($('#moms_md_name').val() && !mom_maiden_name.match(/^(?=.*[a-z]).{3,20}$/)) {
      alert("Mother maiden name not properly entered!");
                    }

blank/empty - work (get result)
disable jquery validation get result
enter wrong input 222/aa - work get (get error msg)
enter correct input - no result at all <-?

else if(!empty($_POST['moms_md_name'])) {

    if (isset($_POST['moms_md_name']) && !preg_match('%^[A-Za-z.' -]{2,30}$%', $_POST['moms_md_name'])) {
        echo "Mother maiden name need to be properly entered!";
        exit();
    }
}