조용한 틈을 타 = 조용한 틈을 타서 = (by) taking advantage of the quiet/lull. (It's an adverbial phrase).
틈 means is a gap, crack, or opening , but it's also widely used to mean an opportunity to do something, an opponent's vulnerability, etc.
- 문틈에 손이 끼었다 = I got my hand stuck in the door jamb.
- 남자애들 틈에 섞여서 노는 여동생 = my sister playing around with boys.
- 죄수들은 도앙갈 틈을 엿보고 있었다 = The prisoners were looking for an opportunity to flee.
- 절친한 사이에 틈이 생겼다/벌어졌다 = A crack crept up in the close friendship.
- 쉴 틈이 없다 = 쉴 사이가 없다 = I can't even take a moment's break.
틈을 타다 (타다 = to ride, go with, or take an opportune chance) means to take advantage of some favorable situation to make one's move, do something sneakily under cover of something, and so on. It's also used as one word 틈타다 when appropriate.
- 도둑은 주인이 한눈파는 틈을 타 물건을 훔쳤다 (틈을 타다) = 도둑은 주인이 한눈파는 것을 틈타 물건을 훔쳤다 (틈타다) = The thief stole the item while the owner was looking away distracted.