Hello,
I don't have an Audio Ape, but looking at the website it looks like it's a really fancy clicker. It definitely will control QLab, but you need to tell Qlab what you want it to do.
Plug it into the mac you're running QLab on, select one of the button presets in Audio Ape that uses MIDI: 
https://audioaperemote.com/pages/tutorials. The one that's called QLab leaves volume up and down as keyboard commands, if you need them.
Now you need to go in to QLab and set what those triggers do. If you just want a go button, go to settings > controls > workspace MIDI. Check the box that says Use "Musical" MIDI Voice Messages. Where it says GO, click the button that says Capture and click the remote button you want to be GO. It should fill in Note on - 2 - 127 or something similar. Repeat this for any of the other controls you want to use (maybe Rewind is Playhead Previous, for example).
If you want a remote button to trigger a specific cue, go to the cue in question. Go to the triggers tab and you'll see a similar MIDI Trigger line. Click Capture, press the control you want to trigger it and it should fill in the details. Make sure the check box is checked.
Worth noting that you've got a 5 button remote, but every button can do 2 things, one on a click and one on a long hold. So you've got 10 possible triggers to play with.
Hope that helps.
Robert