Nice question! First aA comment about terminology. The object you are considering is a "balanced frequency square". Frequency squares generalise Latin squares in the sense that each row and column is a permutation of the same multiset of symbols. The term "balanced" means that each symbol occurs the same number of times.
The question as asked has a negative answer. For example, take b=2 and a to be odd. Let X be the addition table for integers modulo a, and let Y be obtained by adding a to each entry of X. Form F by, initially juxtaposing these blocks<Apolgies:
XY
YX
Now switch the principal entries of these four blocks to make F Some nonsense deleted. For example, when a=3 then F is
312045
120453
201534
045312
453120
534201
Consider an axa submatrix S of this frequency square. Since a is odd,
S must include more than half the rows and more than half of the columns
of one of the blocks. If S misses the principal entry of the block, that means it will include all acost of the symbols that are native to the block, and
that leads easily to the conclusion that S cannot be a Latin square.
If S hitsfiring off an answer on the principal entry it is a little more fiddly, but not too hard to show that S won't be a Latin square.run>
-Ian
