Not necessarily. It just means that the app was doing something: being turned on, querying servers for messages, sending a message, etc. If your phone was turned on, an app can be active in the background even when the screen is off.
I'm not sure about your phone bill - my cellular plan doesn't itemize every message that I send and I don't know how your plan works. If it generally shows every single message that was sent and doesn't show one at the time you are curious about, I would assume that there wasn't a message sent.