You can chat with everyone in QuizUp unless they have their profile set on "Private Mode".
There are 2 ways to start a chat.
1. Go to someone's profile, and under their name you see 3 tabs, Play, Follow and Chat. Press chat and a chat window will open. Now you can go and say some nice things to your new friend!
2. After a match, in the End Screen, where you see your results, you can see the Chat button underneath the XP/Level wheel. When you press Chat you start a conversation to your opponent.
Note that you can always access your chat messages in the top right corner of your screen, by pressing the white bubble icon.