I, too, am an American English native speaker, and I am in complete agreement with Sarah. The word "quiet" can be used as an adjective, an adverb, or a noun, which is why you found "make them be quiet" on Ngram a lot as that is how we use it. The sentence, "Parents should make their children quiet on the train," is definitely one I have never heard, never read, nor one I would ever say. Perhaps in British English, but certainly not in American English. Just my "two cents" worth to break the tie given the differences in opinions here... (*-^)

Thank you Sarah. It's great.

I am an American English speaker, and I would say "Parents should make their children be quiet". I would also say "Can you please ask your child to be quiet?" or complain "Why can't those children just be quiet?" Saying "should make their children be quiet" is expressing it like a command that the parents should be giving the children -- like "Be nice", "be good", "be quiet", "behave!" However, "Parents make their children be quiet on the train." sounds rude, like a complaint. It would be more polite to say "Parents should keep their children quiet on the train."

Thank you Rena. I find "make them be quiet" on Ngram fairly a lot. When do you think it's used?

make someone do something make someone pay you for your work make your sister stop hitting you. ---------- make someone adjective make someone happy You've made me very happy today. That news makes me sad. Your "Parents should make their children quiet on the train" quiet is an adjective. I think "make someone BE (+ adj)" is weird. I would not say that.