Nice question! It is a symbol in several ways, and of several things. First and most simply, it indicates how quiet the room is. In that instance, it is barely a symbol, but rather just a sign. However, it becomes a symbol, representing more than just the quietness or Emily's mode of dress, when put into context. The silence comes from the respect the town has for her, and so the very quiet is symbolic. The watch is on a gold chain, just as Emily is "chained" by her father's wealth. It, and the time it counts off, moves on, even when no one can see it passing; this will be symbolic of the rest of story to come, especially of how time passes regarding the town and Emily's beau. This passage of time is at the end of the chain, just as things are at the end of their ropes (metaphorically). When time passes, Miss Emily ends up at the end of her rope.
To further add to the symbolism, look at the other instances of gold in that area of the story—the tarnished can head, the tarnished gilt easel. Golden (precious) things are tainted in this house.