I don't think so, write the moves down yourself!
Little ChessPartner Game Score
Just open Winboard and as you make your moves in LCP make them in Winboard at the same time and save the game.
Seems like you should be able to export form LCP a .txt or .pgn or some other file of the completed game though. Just needs somebody with access to the full source code in order to add functionality. Is it open source? Not especially sophisticated so no real need to be proprietary.
Is there a way to get the moves of a game vs the computer? I almost beat it, and I would like to study the game.