If you are talking about the past, the standard, modern English form would be 'Didn't they have any children?'. 'Had they no children?' is technically correct, but very unusual. It sounds strange and archaic to our modern ears, although it may be used in some regional dialects.

Hi It is not correct to use this word/tense in this format. It can be said in a statement such as this - "had they no children, they could travel the world" but not like what you've mentioned where she has only used it to form a question. It should be structured like this - "do they not have children?" Or "have they no children?".

American English: It's not incorrect, but it's very weird. It's a bit old fashioned. I would be much more likely to use "Didn't they have any children?" than "Had they no children?" Also, that would not be asked of people who are still alive. (For them, you would use "Don't they have children?" or "Have they no children?" or "Haven't they any children?") If you say "Had they no children?" you're asking about dead people.

"Had they no children?" is technically correct, but certainly not standard. It sounds like 19th-century English, or from the British countryside. This is one of the throwbacks to our Germanic grammar, which is why it sounds a bit archaic. In the upper levels of an English course, you would come across this phrasing - as a colloquial question, or even in the inverted conditional that Shakira provided.