Also, how can I test transposition tables to check that the code I added with regards to TT works fine?

One thing I've done is to perform each search with TT on and TT off and then check that the same result is being returned. The TT shouldn't affect the result (only in extremely rare instances) compared to a normal search - it should only affect the speed.

Unfortunately the result is affected much more often than "in extremely rare instances". Search instability is a well-known phenomenon in alpha-beta search, and transposition tables are responsible for quite a bit of it. In chess this is particularly easy to trigger in situations where draw by repetition or the 50-move rule are involved.

http://chessprogramming.wikispaces.com/Search+Instability

It appears that most of these are caused by interactions with other search enhancements. Surely for the purpose of proving that an implementation is correct, you would disable these enhancements initially?

A lot of it also has to do with graph-history interaction. It is possible that nine men morris doesn't have any such issues, and perhaps then this isn't much of an issue.

What happens when a position is repeated in nine men morris? Is the game a draw? If that's the case, there is the potential for trouble already.

When adding quiescence search with TT, should the code be similar to this when depth == maxdepth??

        if (depth == maxDepth)
        {
        	val = quiesce(board, alpha, beta);
        	setTT(key, depth, val, TTEntry.BOUND_EXACT, null);
        	
        	return val;
        }

Is it also necessary to save the move struct in the TT? Can it be used for something??

I would store the position in the transposition table inside `quiesce'. Storing the move can be useful, because later on you are likely to visit this position again at a higher depth, and the move that turned out to be best in quiescence search is a reasonable candidate to try first.

Disclaimer: I have never obtained an improvement in my programs by using transposition tables in the quiescence search, so I don't use it.

Well then I will not need quiescence search with TT :)