Well, I think you have the following possibilities:
1) Have you installed a serial mouse in your current computer and does it work? If not, you might need a serial mouse driver installed. It is possible (but unlikely) that the Atari TB will work once the proper driver is added to the system.
2) It is possible that the trackball just has some wires out of place from a standard serial connection, so you could maybe re-arrange the wires or cut off the serial connector and run the wires to a terminal strip and connect them differently, put this is only slightly less work than a mouse hack.
3) Interfacing, See the optical interfaces part of
my chart, basically you have the following three options:
Mouse hack - See
Note 49 on my page:
Opti-Pac or
Mini-Pac from
www.ultimarc.com - basically a pre-built mouse hack.
Also the
Opti-Wiz from
www.groovygamegear.com - also basically a pre-built mouse hack.