The noun intimation means a hint or an indirect suggestion. Your teacher's intimation that there could be a quiz the next day might send you into a panic, while your friend sitting beside you might not even notice.
Intimation comes from the Latin word intimationem, which means an announcement. In English, intimation refers to a less direct form of communication. It's a suggestion or hint, rather than a blatant statement of fact. Your first intimation that your brother had a girlfriend was the amount of time he spent whispering into the phone. The second intimation was when he asked your parents for money for two movie tickets.