How to perform a Stockwell Transform on a 2D matrix
There is a MATLAB code available for the Stockwell transform here. It also includes a code for the one-dimensional transform but it is less efficient that the Mathworks file exchange code that is mentioned in the question. This is because one uses for loops and the other doesn't.

Using this 2D S-transform function, you can quickly run into memory issues. You can easily modify the code to specify the exact positions of x and y that you need, so that S is not 4 dimensional.