Warning - I am a belt-and-suspenders-type person. You could probably do this in less steps, but I like to double-check everything....
Copy your COH folder to get all the Issue 24 files and put them in a place you will remember (lets say C:\COHFILES)
Then start the PChat client (but don't go into COH). Clicky the picture of the folder and copy all the files in that folder into the new PC (say, C:\COHOLD) .
Install PChat on the new PC. Use the folder option to find the location of the new folder. Copy the files from C:\COHOLD into that folder.
(If PChat doesn't find the Issue 24 files, use Browse and click C:\COHFILES and it will locate them)
Just to make sure we don't overwrite some location-specific files, install PChat again after the copy. Start PChat, make sure it works, then install QChat.