While I don’t have any direct experience with the CQ options, my gut feeling is the answer is “No”.
I have used lots of other A&H digital consoles and I’ve never seen one that interfaced directly with MIDI devices correctly. You will likely need some sort of interface between the CQ and the Nexfoot device to interpret the data flowing between the two.
If you want a hardware device (something that is literal plug and play when setting up for an event), the the Bomebox is available. Unfortunately it is pretty expensive, especially for such a “small need” as this. Another alternative is the third party app Mixing Station (requires a license to use the MIDI functions, but it is very inexpensive). You can plug the Nexfoot directly into the Mixing Station host (spare mobile phone or a tablet perhaps) and start the Mixing Station software. But obviously that has a lot more “moving parts” and things that could go wrong that something like the Bomebox.
Looking at the NEXFOOT specifications, this seems theoretically possible, but I think there are a couple of catches
The CQ supports MIDI control over USB-B, including channel levels, mutes, and mute groups. The NEXFOOT also has a USB-A host port for connecting to USB MIDI devices.
However, having compatible USB connections doesn’t necessarily mean the devices can communicate properly. The CQ uses NRPN messages for native MIDI control, so the NEXFOOT would need to support sending those specific commands.
So even though the hardware and MIDI capabilities suggest it might work, personally I wouldn’t recommend purchasing it specifically for CQ control without confirmation from LIVTRA that it supports NRPN. There are also a couple alternatives that you could look into as well like Mixing Station.
Another option too is using a MIDI foot controller through a laptop running Allen & Heath MIDI Control in CC Translator mode. I know that’s a way to translate standard MIDI commands into the CQ’s NRPN message protocol.