You said you're making your game under Qt, however you never specified or mentioned any graphical rendering engine, I assume you're not using OpenGL or any third party graphical rendering engine, or further more a third party Game Engine.
So, what I can think of when creating a console, is to maybe add a ListWidget (not ListView unless you know how to work with models, etc., etc.) and a textfield and a pushbutton.
So the structure for the input/output console would be like so (pardon my poor drawing skills):
So ultimately you will use the ListWidget to handle all the textual game information and even the information sent by the player, etc., etc. the QPushButton will serve as the actual event that will fire off the "trigger" that player has actually sent something, the TextEdit will simply act as the placeholder for player "commands" to send to the game.
Good Luck!
Cheers!
Edit
Didn't noticed this was questioned a month ago! My apologies if I caused necro-posting! just wanted to help!