I'm trying to write a chess game and find that I cannot find solutions to find a stalemate situation. I'm trying to google, but can't find anything. Is there a well-known algorithm or something?
Your move generator will be one of two different designs;
The former is better as it doesn't need post-processing.
A stalemate condition is simply one where there are no legal moves and the moving-side's king is not in check. A checkmate condition is one where there are no legal moves but the moving-side's king is in check.
In other words if you've figured out how to detect check and checkmate, you've already got everything necessary to detect stalemate.