To cross-question someone is to ask them many careful, detailed questions, often one after another, to check their story. It is close to “cross-examine,” the courtroom word, but you can use it outside court: a parent, a reporter, or a teacher might do it. The asker is not making small talk; they want answers that fit. You will meet the verb more than the noun. It sounds a little old-fashioned and more British than American; many people would now say “grill” or “question closely.” It is stronger than a normal “ask.”