From this some obvious patterns emerged

1. m = 2

   1. S(2, 2) = 5

   2. S(2, 3) = 9 (+4)

   3. S(2, 4) = 15 (+6)

   4. S(2, 5) = 21 (+6)

   5. S(2, 6) = 27 (+6)

2. m = 3

   1. S(3, 3) = 13

   2. S(3, 4) = 17 (+4)

   3. S(3, 5) = 23 (+6)

   4. S(3, 6) = 29 (+6)

   5. S(3, 7) = 35 (+6)

   6. S(3, 8) = 41 (+6)

3. m = 4

   1. S(4, 4) = 21

   2. S(4, 5) = 25 (+4)

   3. S(4, 6) = 31 (+6)

   4. S(4, 7) = 37 (+6)

4. m = 5

   1. S(5, 5) = 29

   2. S(5, 6) = 33 (+4)

   3. S(5, 7) = 39 (+6)

Essentially, we have the following:

This was not surprising as I anticipated a recursive solution since after some moves a board can essentially result in a smaller board (cutting off the useless parts which we will no longer touch).

From here, you can either notice more patterns or count solutions manually in many ways, I went the following path to get the answer. Take note of this page: https://proofwiki.org/wiki/Square_Modulo_8 to understand the first line.