In your EverQuest install folder (where eqgame.exe resides), there is also a file named eqclient.ini. Open this file. (If you can't find it, you aren't looking hard enough. The game won't run without it.)
This file has many values in it. Look for the following value:
(line 12 for me)
And change to:
Exit the game (if running) and re-launch, and log into a character. You will now see a log file under the Logs folder of your EverQuest install, and will record all of your chat windows until you disable it.
Hope this helps!