Translation:They don't have daughters.
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It's an unusual thing to say, but it's not out of the question.
"Have got" is weird for American English, but normal for British. And for "They have not got daughters" you'd need a very specific context, like where you're emphasizing the assertion to somebody who assumes they do have daughters.
Depending on the discourse context, dialect, speech register, and so on, people could say "They don't have a daughter," "They don't have daughters," "They don't have any daughters," "They haven't a daughter," "They have no daughter," "They haven't got a daughter," "They have got zero daughters," and lots of other variations.
"They don't have a daughter" would be correct for the singular (as you say the Chinese doesn't indicate how many) The reason "any daughter" doesn't sound natural is because "any" is used in front of nouns to refer to indefinite or unknown quantities; all countable nouns (e.g. apples, chairs, daughters...) are in plural and all uncountable nouns (e.g. money, bread, etc.) are in the "singular".
"They have no daughters" equals "They don't have daughters" and should be accepted. The main solution is incorrect English, that is "They have not got daughters." It is not slang,, but wrong usage, and therefore, the correct solution should be "They have no daughters" or secondly "They don't have daughters.
Asked a China native why "They do not have a daughter" is incorrect.
You have to look at that as "any daughters" because there is no qualifier. Only if the noun exists would you use a qualifier (as in: "There are 2 girls standing there" or "there is a girl standing there"). If there is a "non-existence" you would simply use the translation of "any NOUNs" (as in "there are no girls standing there" instead of "there is not a girl standing there").