How to retrieve permanently deleted files from my old macbook hdd drive by using my new macbook laptop?

How to retrieve permanently deleted files from my old macbook hdd drive by using my new macbook laptop?

I can open my old macbook hdd as an external drive with my new macbook air , however the files I need are not there and I think it got deleted or corrupted. Is it possible to still retrieve it ? How about using time machine in my new macbook? Can it restore my old hdd's deleted files?

Thank you.