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You can access the $(i,j)$-th block using B.subdivision(i,j).
For your more general problem:
sage: A = random_matrix(ZZ,12,12)
sage: A.subdivide([3,6,9],[3,6,9])
sage: B = block_matrix([[A.subdivision(i,j) + A.subdivision(j,i) for i in range(4)] for j in range(4)])
sage: A
[ 0 -1 1| 1 0 1| -18 -1 -2| -1 1 -1]
[ -1 8 -312| 16 1 -3| 3 0 0| 0 -1 0]
[ 0 -13 -1| 4 2 -1| -1 1 1| 21 -6 1]
[--------------+--------------+--------------+--------------]
[ 1 -3 -4| 2 -2 2| -3 2 0| -1 3 0]
[ -1 1 1| -2 -1 -2| -1 -3 -1| 0 -1 6]
[ 5 1 1| -1 -1 -2| -7 -1 0| 1 -1 1]
[--------------+--------------+--------------+--------------]
[ -1 4 -3| 1 4 -15| 5 1 1| 1 0 2]
[ 5 0 0| 2 -1 -1| -16 -2 1| 0 -2 -1]
[ 0 1 0| -2 1 5| 8 -11 1| -7 -1 -2]
[--------------+--------------+--------------+--------------]
[ 1 1 -4| -6 -1 3|-217 -1 0| 3 0 -1]
[ -1 0 1| 0 -7 0| -1 3 1| -3 3 1]
[ -1 0 0| 2 0 -1| -2 -2 0| -1 -2 0]
sage: B
[ 0 -2 2| 2 -3 -3| -19 3 -5| 0 2 -5]
[ -2 16 -624| 15 2 -2| 8 0 0| -1 -1 1]
[ 0 -26 -2| 9 3 0| -1 2 1| 20 -6 1]
[--------------+--------------+--------------+--------------]
[ 2 -3 -3| 4 -4 4| -2 6 -15| -7 2 3]
[ 15 2 -2| -4 -2 -4| 1 -4 -2| 0 -8 6]
[ 9 3 0| -2 -2 -4| -9 0 5| 3 -1 0]
[--------------+--------------+--------------+--------------]
[ -19 3 -5| -2 6 -15| 10 2 2|-216 -1 2]
[ 8 0 0| 1 -4 -2| -32 -4 2| -1 1 0]
[ -1 2 1| -9 0 5| 16 -22 2| -9 -3 -2]
[--------------+--------------+--------------+--------------]
[ 0 2 -5| -7 2 3|-216 -1 2| 6 0 -2]
[ -1 -1 1| 0 -8 6| -1 1 0| -6 6 2]
[ 20 -6 1| 3 -1 0| -9 -3 -2| -2 -4 0]
