In order to perform a nandroid backup you need to do this from a custom recovery. In order for you to get a custom recovery on your phone you must first Root it.
You can backup all your downloaded apps to your uSD card with AppTyrant. Transfer the contents of the uSD card to a folder on a PC.
Google offers some saving of phone content, but I'm not sure if it is anything that is not already saved to the uSD card.
Until you root that's the best you can do.
There used to be a utility that would establish a Shell Root then would allow you to put a custom recovery on a phone before you did the permanent root (ROMTools), but it is a dead thread with dead links now and used SuperOneClick to establish the temp root. ZV4 also allows a temp root, then you would have to install the recovery from a command prompt or terminal emulator, but I don't think it will work on the Q either.
For your phone it's root, then recovery, then backup.