Quiet: You make a sound but not a loud sound.
✗ I have a car with silent engine. (silent engine = broken car)
✓ I have a car with quiet engine. (all cars make sound, however this car doesn't make a loud sound -> quiet)
Silent: You don't make any sound at all (no noise, no sound)
✗ She didn't say anything, she was quiet.
✓ She didn't say anything, she was silent.