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Simply use Part and Set:

f[A_?SquareMatrixQ, i_Integer, j_Integer] := Module[{p, q, B},   q = Range @@ Sort[{i, j}]; p = RotateLeft[q, Sign[j-i]];   B = A; B[[q]] = B[[p]];   B[[All, q]] = B[[All, p]]; B   ] A = Outer[Plus, Range[5], Range[0, 4]]; A // MatrixForm 

$\left( \begin{array}{ccccc} 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & 4 & 5 & 6 \\ 3 & 4 & 5 & 6 & 7 \\ 4 & 5 & 6 & 7 & 8 \\ 5 & 6 & 7 & 8 & 9 \\ \end{array} \right)$

B = f[A,2,4]; B // MatrixForm 

$\left( \begin{array}{ccccc} 1 & 3 & 4 & 2 & 5 \\ 3 & 5 & 6 & 4 & 7 \\ 4 & 6 & 7 & 5 & 8 \\ 2 & 4 & 5 & 3 & 6 \\ 5 & 7 & 8 & 6 & 9 \\ \end{array} \right)$

Simply use Part and Set:

f[A_?SquareMatrixQ, i_Integer, j_Integer] := Module[{p, idx}, idx = Range[i, j, Sign[j - i]]; p = Range[1, Length[A]]; p[[idx]] = RotateLeft[idx]; A[[p, p]] ] A = Outer[Plus, Range[5], Range[0, 4]]; A // MatrixForm 

$\left( \begin{array}{ccccc} 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & 4 & 5 & 6 \\ 3 & 4 & 5 & 6 & 7 \\ 4 & 5 & 6 & 7 & 8 \\ 5 & 6 & 7 & 8 & 9 \\ \end{array} \right)$

B = f[A,2,4]; B // MatrixForm 

$\left( \begin{array}{ccccc} 1 & 3 & 4 & 2 & 5 \\ 3 & 5 & 6 & 4 & 7 \\ 4 & 6 & 7 & 5 & 8 \\ 2 & 4 & 5 & 3 & 6 \\ 5 & 7 & 8 & 6 & 9 \\ \end{array} \right)$

Simply use Part and Set:

f[A_?SquareMatrixQ, i_Integer, j_Integer] := Module[{p, q, B}, q = Range @@ Sort[{i, j}]; p = RotateLeft[q, -Sign[i - j]]; B = A; B[[q]] = B[[p]]; B[[All, q]] = B[[All, p]]; B ] A = Outer[Plus, Range[5], Range[0, 4]]; A // MatrixForm 

$\left( \begin{array}{ccccc} 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & 4 & 5 & 6 \\ 3 & 4 & 5 & 6 & 7 \\ 4 & 5 & 6 & 7 & 8 \\ 5 & 6 & 7 & 8 & 9 \\ \end{array} \right)$

B = f[A,2,4]; B // MatrixForm 

$\left( \begin{array}{ccccc} 1 & 3 & 4 & 2 & 5 \\ 3 & 5 & 6 & 4 & 7 \\ 4 & 6 & 7 & 5 & 8 \\ 2 & 4 & 5 & 3 & 6 \\ 5 & 7 & 8 & 6 & 9 \\ \end{array} \right)$

Simply use Part and Set:

f[A_?SquareMatrixQ, i_Integer, j_Integer] := Module[{p, q, B}, q = Range @@ Sort[{i, j}]; p = RotateLeft[q, Sign[j-i]]; B = A; B[[q]] = B[[p]]; B[[All, q]] = B[[All, p]]; B ] A = Outer[Plus, Range[5], Range[0, 4]]; A // MatrixForm 

$\left( \begin{array}{ccccc} 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & 4 & 5 & 6 \\ 3 & 4 & 5 & 6 & 7 \\ 4 & 5 & 6 & 7 & 8 \\ 5 & 6 & 7 & 8 & 9 \\ \end{array} \right)$

B = f[A,2,4]; B // MatrixForm 

$\left( \begin{array}{ccccc} 1 & 3 & 4 & 2 & 5 \\ 3 & 5 & 6 & 4 & 7 \\ 4 & 6 & 7 & 5 & 8 \\ 2 & 4 & 5 & 3 & 6 \\ 5 & 7 & 8 & 6 & 9 \\ \end{array} \right)$

Simply use Part and Set:

f[A_?SquareMatrixQ, i_Integer, j_Integer] :=  Moduel[{B}, B = A;  B[[{i, j}]] = B[[{j, i}]]; B[[All, {u, j}]] = B[[All, {j, i}]]; B ] A = Outer[Plus, Range[5], Range[0, 4]]; A // MatrixForm 

$\left( \begin{array}{ccccc} 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & 4 & 5 & 6 \\ 3 & 4 & 5 & 6 & 7 \\ 4 & 5 & 6 & 7 & 8 \\ 5 & 6 & 7 & 8 & 9 \\ \end{array} \right)$

B = f[A,2,4]; B // MatrixForm 

$\left( \begin{array}{ccccc} 1 & 4 & 3 & 2 & 5 \\ 4 & 7 & 6 & 5 & 8 \\ 3 & 6 & 5 & 4 & 7 \\ 2 & 5 & 4 & 3 & 6 \\ 5 & 8 & 7 & 6 & 9 \\ \end{array} \right)$

Simply use Part and Set:

f[A_?SquareMatrixQ, i_Integer, j_Integer] := Module[{p, q, B}, q = Range @@ Sort[{i, j}];  p = RotateLeft[q, -Sign[i - j]]; B = A; B[[q]] = B[[p]]; B[[All, q]] = B[[All, p]]; B ] A = Outer[Plus, Range[5], Range[0, 4]]; A // MatrixForm 

$\left( \begin{array}{ccccc} 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & 4 & 5 & 6 \\ 3 & 4 & 5 & 6 & 7 \\ 4 & 5 & 6 & 7 & 8 \\ 5 & 6 & 7 & 8 & 9 \\ \end{array} \right)$

B = f[A,2,4]; B // MatrixForm 

$\left( \begin{array}{ccccc} 1 & 3 & 4 & 2 & 5 \\ 3 & 5 & 6 & 4 & 7 \\ 4 & 6 & 7 & 5 & 8 \\ 2 & 4 & 5 & 3 & 6 \\ 5 & 7 & 8 & 6 & 9 \\ \end{array} \right)$