If you're using variables for key names, then no, don't quote them.
If you're just referencing an associative array using a string, then yes, you must quote them, i.e.
$data = array();
$data['key_name'] = 'hi';
The only time you don't need to quote the key name is if you're specifically trying to use a constant, which I doubt 99%-100% if the people that don't quote their keynames are doing. This is such an example:
define('this_is_a_constant', 'key1');
$data = array();
$data['key1'] = 'Hello World!';
echo $data[this_is_a_constant]; // Hello World!