They're two different vowels and are pronounced dirrefently. Finnish doesn't mix A and Ä in the same word very often, because these vowels are formed in a different part of the mouth and therefore would feel unnatural to pronounce. Same goes for O and Ö.
Exceptions tend to be either loanwords from other languages or compound words. For example jääkaappi (fridge) is okay because it's a compound word of jää (ice) and kaappi (cupboard/closet).
I don't know where you're getting the "quiet" from, but "hiljaa" is an adverb, not an adjective. Also, "hiljat" is not a word (although it is the nominative plural form of a person's name). The nominative singular form of the adjective is "hiljainen". That sentence requires partitive plural for the noun phrase though, so it should be "heillä on hiljaisia nättejä lehmiä". This is because countable nouns in ownership clauses and existential clauses require partitive plural when the noun has several referents but their number is unspecified.