Prueba:
$notices = DB::table('notices')
->join('users', 'notices.user_id', '=', 'users.id')
->join('departments', 'users.dpt_id', '=', 'departments.id')
->select('notices.id', 'notices.title', 'notices.body', 'notices.created_at', 'notices.updated_at', 'users.name', 'departments.department_name')
->paginate(20);