Once the window where the chat was taking place has been closed, the only way to get at the information is through the logs, presuming you did log.

Without knowing what format you've got setup on your system for storing the logs..ie: where the logs are stored and how the names are created, there's nothing more I can do at this point.

To view logs from within mIRC, click Tools - Log Files, which will bring up a dialog where you can search for a particular log