Extended Domino Dots
Pyth, 37 bytes

jc:*d*QQiRQhos.aMNf.Am}_dTT.c^UQ2vzNQ


Slow solution. Generates all possible pairs of [0, m), generate all possible n-combinations of those pairs (without repetition). Filter each combination of pairs such that if P is in the combination, so is reverse(P). This guarantees diagonal symmetry. Sort by the sum of distances to the origin. Decode the pairs of numbers as base-m numbers, giving a list of indices. Create a string of spaces except for those indices, putting quotation marks instead. Finally chop into m pieces and join by newlines. Creates outputs like such:

""""
"""
""
"