<form  class="form-inline" method="post" >
     <div id="div_id_value" class="form-group">

        <input  class="numberinput form-control"
          id="value" name="value"
          placeholder="Value (mmol/L)"
          required="True" type="number" step="any" min="0"/>
        </div>


      <div id="div_id_category" class="form-group">

          <select  id="id_category" name="category">
            <option value="Breakfast">Breakfast</option>
             <option value="Lunch" >Lunch</option>
              <option value="Dinner">Dinner</option>
              <option value="Snack">Snack</option>
              <option value="Bedtime">Bedtime</option>
               <option value="No Category" selected="selected">No Category</option>
             </select>
           </div>

           <input  type="submit" name="submit" value="Quick Add" id="quick">
       </form>

<script src="https://ajax.googleapis.com/ajax/libs/jquery/3.3.1/jquery.min.js">


  $("#quick").click(function() {

      var sugar2 = $("#value").val();
      var category2 = $("#category").val();

      $.ajax({
        type:'POST',
        url:'quick.php',
        data:{ 'sugar': sugar2, 'category':category2},
        dataType:'json',
        success: function(output) {
    alert(output);
         };
      });
  });

<?php
  session_start();
  include 'conn.php';

  if (isset($_POST['sugar'])) {

    $sugar = $_POST['sugar'];
    $category =  $_POST['category'];
    $email= $_SESSION['email'];

    $query = "INSERT INTO data (email, sugar, category)
            VALUES($email, $sugar, $category )";


   if(mysqli_query($link, $query)) {

                header("Location: index.php");


          };
         };
        ?>

$("#quick").submit(function(evt) { ... });
//or
$("#quick").on('submit', function(evt) { ... });

<?php
session_start();
include 'conn.php';

if (isset($_POST['sugar'])) {

  $sugar = $_POST['sugar'];
  $category =  $_POST['category'];
  $email= $_SESSION['email'];

  $query = "INSERT INTO data (email, sugar, category)
            VALUES($email, $sugar, $category )";

  //save the output of the result
  $result = mysqli_query($link, $query);

  //according to the docs, mysqli_query() will return false on failure
  //check for false

  if( $result === false ) {

    //the query failed
    //provide a response to the AJAX success function
    echo "Query failed. Check the logs to understand why, or try enabling error reporting.";

  }else {

    //the query worked!
    echo 'All good!'; //provide a response to the AJAX success function

  }

  //Kill PHP. Good idea with an AJAX request.
  exit;
}

<?php
  echo "my response";
?>