Differences between no child vs no children
Swan in Practical English Usage (p352) has a good entry on this question:
After no, countable nouns are usually plural unless the sense makes a singular noun necessary. Compare:
He's got no children. (More natural than He's got no child.)
He's got no wife. (More normal than He's got no wives.)
So, They have no children is indeed more common and natural than They have no child. Conversely, They have no car seems more natural than They have no cars.