Do I really have to tell my manager another employee is quitting work?
Is that person going to quit without letting anyone know?
That's what he said. I just don't think it's my responsibility to let my manager know, though.
I think you should let someone know, just to protect yourself. They might find out that you knew about his wanting to quit and be upset that you didn't say anything. It might not be your responsibility, but keeping records at work is always important.
Nah, I'm about to leave. I have other more important things to do.
I understand, but if you know that the employee is quitting you should tell your manager so he can make other arrangements. If the other person quits and there is no one available to take his place then it could cause a big issue for everyone.