Calm is mostly used to describe an emotional state of tranquility, relaxation, and without any emotional disturbance or instability. It is also used to describe the sea when there are no waves, or the calm after the storm. I felt quite calm before my exam. The sea is very calm today, so you can swim.
Quiet, on the other hand means "without noise" or "with few people/with little activity". The room was quiet when I walked in (without noise). The underground was very quiet this morning (with few people). I think I'll just have a quiet night in tonight (with little activity).