Simply use `Part`:

 f[A_?SquareMatrixQ, i_Integer, j_Integer] := 
 With[{p = PermutationList[Cycles[{{i, j}}], Length[A]]},
 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

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