There seems to be some confusion in your post between lighting cues in QLab, and using QLab to control external lighting boards.
Light Cues in QLab were introduced in QLab 4 and control lights directly through dmx or artnet interfaces.
QLab has always been able to control external lighting boards through MIDI cues, which send MIDI. MIDI cues can send MSC ( MIDI show control) which refers to your Tiger Touch Cue list by deviceID and cuelist ID, which you set up as described in the manual linked below. You can also send MIDI voice messages like note ons or offs, or cc values which the Tiger touch can map to its buttons and faders. I would start by getting the MSC command GO to trigger a cue list on your Tiger Touch. Once you set the device ID and cue list ID on your Tiger Touch that should be straightforward.
Avolites Titan Series consoles, including Tiger Touch, do not have OSC implemented.
They can only be controlled with MIDI. Either using a the 5 pin DIN MIDI in on a standard Tiger Touch or additionally, USB on a II or Pro version.
Your MIDI beam just replaces a MIDI 5 pin cable, so you still need a physical MIDI interface on the Mac running QLab to connect to the MIDI in of the MidiBeam transmitter.
I would get MSC from QLab working using an interface and a cable first, then try to replace the cable with the MidiBeam..