No. Because the best PS5 controllers, such as the DualSense Wireless controller, have been created to make use of 'next-gen games' with high-spec features such as haptic feedback and adaptive triggers, these controllers won't work on old consoles.
You can however use a PlayStation 4 controller with your PS5, but the catch is that it can only be used to play PS4 games and won't work on PS5 titles.