Even in the simple case of F_2 matrices, I'm not sure it is easy to do better than brute force. But no, you don't need to write out each matrix product. This seems to work.
m = 3;
% all possible pairs of 2x2 matrices, as base m integers, doubles
[Adoub,Bdoub] = meshgrid(0:m^4-1);
% convert those "matrices" into pages of 2x2 matrices,
% with each matrix as a plane in a 3-d matrix
Am = reshape(dec2base(Adoub,m)' - '0',[2 2 m^8]);
Bm = reshape(dec2base(Bdoub,m)' - '0',[2 2 m^8]);
% just take the product of all combinations of matrices, in F_m
AB = mod(pagemtimes(Am,Bm),m);
% and test for all zeros
zeroind = all(reshape(AB,[4,m^8]) == 0,1);
table(dec2base(Adoub(zeroind),m),dec2base(Bdoub(zeroind),m))
There are 58 sets of matrices identified for base 2, and 417 for base 3. If m is too large of course, expect things to get nasty in terms of computations, because you will have m^8 such combinations of matrices to deal with.
