Обычный SQL,
$sql = "SELECT `id`, `first_name`, `last_name`, `mobile`, CONCAT(`first_name`, ' ',`last_name`) AS `full_name` FROM `users`
WHERE `first_name` LIKE '%".$word."%'
OR `last_name` LIKE '%".$word."%'
OR CONCAT(`first_name`, ' ',`last_name`) LIKE '%".$word."%'
OR `mobile` LIKE '%".$word."%';";
В Laravel,
$result = User::select('id', 'first_name', 'last_name', 'mobile', DB::raw('CONCAT(first_Name, " ", last_Name) AS name'))
->where('first_name', 'LIKE', '%'.$word.'%')
->orWhere('last_name', 'LIKE', '%'.$word.'%')
->orWhere(CONCAT(first_Name, " ", last_Name), 'LIKE', '%'.$word.'%')
->get();