you are right but what you mean ./? A quick child was born to them. This dress sit ill on you

The first is fine. The second is technically correct but not recognized to use the adjective "quick" in such context (with the meaning "live"). I would suggest, "A live child was born to them". The third I would suggest adding "up" as "Yesterday I had to sit up late" if the meaning is about staying awake late into the night. The fourth is not correct at all and could be said instead as "The dress is not your style" or something like that.

does it means a quick child as in, an intelligent child? I would change the last one to That dress does not sit well on you, but it is a kind of odd thing to say. A more natural expression would be "that dress doesn't suit you"

A quick child was born to them sounds awkward. More contextual evidence is needed in order to make this sentence logical. There is not a problem grammatically, the error lies with semantics.