Here is a fast example using get_categories
instead of a plugin or the walker class, maybe it will help you in the right direction, it will list all children of a parent category in json format. As brasofilo mentions since you require a specific format you will want to build a custom array.
// let's get the bare minimum going
$args = array(
'type' => 'post',
'child_of' => 20, //change this, hardcoded as example
'taxonomy' => 'category'
);
$categories = get_categories( $args );
$json = json_encode($categories);
var_dump($json); //do what you want