I have the Droid Turbo, and Verizon Messages + 6.2.3 - when a text, or email comes in, I DO get the convo bubble, or an envelope, on the notification screen when I wave my hand over the phone. Sometimes the text indicator will get bumped off the screen due to other notifications, but it does pop up.
I don't have any special settings that I am aware of, this is the way it has always worked o the Turbo for me. I wish I could be more helpful!