You can read about ajax in javascript and make an ajax request that gets the json object from PHP then process that in your javascript:

Example (with some improvement to your PHP code)

PHP
  
$chart6 = $controller->runQuery("SELECT DATE(pay_date) `DATE`, SUM(amount) daily_amount, DATE_FORMAT(pay_date,'%a') DAYNAME FROM tbl_paid WHERE DATE(pay_date) BETWEEN DATE(:today) - INTERVAL 1 WEEK AND DATE(:today) GROUP BY DATE(pay_date);"); 
$chart6->execute(array(":today"=>$today));
$labels = array();
$series = array()
while($fetch = $stmt->fetch(PDO::FETCH_ASSOC)) {
  $labels[] = $fetch['DAYNAME'];
  $series0[] = $fetch['daily_amount'];
}
$series[] = $series0;

// add the labels and series to an array, then convert that to json
$data = array('labels' => $labels, 'series' => $series);
echo json_encode($data);

JS:

// Make an Ajax request, here I am using jQuery $.getJSON  but you can use $.get or $.post if you need to have it as a post request, or even $.ajax

$.getJSON('https://example.com/database_fetch_page.php', function(data) {
  new Chartist.Line('#headline-chart', data, options);
});