First, I would not invest in any product to integrate into a 17 year old matrix. inform the customer this must be retired. I bet they have some burned in CRT monitors.
Encode the existing fixed cameras with M7016 encoders (1000 bucks each). I assume the existing PTZ analog use UTP (not up the coax control). The axis encoders should support this. If not, Axis has some more expensive encoders with up the coax control.
Having Genetec already with base license is good. With 120 analog and 20 new IP PTZ cameras, probably need a new server, unless VM/SAN based.
Get a i7 workstation with Quad card. Two large screens 64 cameras each and two drop downs with a axis keyboard.
To start cheaper, get the server that will eventually be needed, but add video card. Add Axis joystck or monitor rs485 bus from vicon for control of the genetec server/workstation to PTZ. Will probably be latency. Will display the 20 PTZ cameras on maybe a dual monitor setup. Call Genetec to see if Vicon control can be implemented/relayed. I believe Axis may have wired rs-485 which will need extra network UTP to camera. Will not make sense if the ip cameras go to remote it closets.
I would bet the matrix works with some multiplexors to view 4x4 video, but with this many cameras, I am sure they only show a fraction live. I also bet that they are not recording all 120 cameras, so a full upgrade may be a sell from this to fully migrate to ip/pc control.