I think considering how a descriptor word like 「静か」 could operate as a target would be helpful to consider. This word is not a person, place, thing, idea etc. Instead, it is a descriptor word. I would call 静か a な-adj or an adjectival noun, so when に is used is is always pretty safe to say it is used as an adverb vs. a target of an action. You can't give a letter to "quiet", you can't go to "quiet", you can't ask a question to "quiet."
Now for 静かになった I guess you could see it as both a adverb and a target, but not in the same sense as the sentence you presented about a letter. This grammar construct uses the form ~になった if ~ is a な-adj or a noun, but uses ~くなった where ~く is the adverb form of an い-adj (aka adjectival verb). So while the English translation "Everyone became quiet" doesn't use "quietly" I would say it is safe to see 静かに as an adverb.