Main thing for a quiet rider is soft hands and they never use aids incorrectly. My trainer describes me as a quiet rider so I get handed all the hyper horses because they tend to calm down when they have quiet rider.
A quiet jumper is one that doesn't rush at jumps or pull on their riders hand, carries themself nicely, and jumps naturally with ease.
A quiet rider isn't "noisy" and all over the place with their aids, doesn't interfere with the horse, and is very educated in how they use their aids, not being harsh and jerkey with their reins, and not kicking excessively hard. Quiet riders also follow their horse very well, and are usually a nice picture to watch, but they don't have to be.
A quiet horse is a horse that doesn't spook and goes over his jumps with ease; the horse might even be a little bit lazy.
A quiet rider means that they use subtle but gentle aids with their legs and hands to connect with their horse. A lot of grand prix jumpers are very soft and with their rein aids. They can keep their hands in contact with the horse without moving them a lot.