The answer is, in general, no. Here’s why. The eigen values of A are the square of the eigen values of A^2. Put  in diagonal form with the eigen values down the main diagonal. Square any diagonal matrix, and the values on the diagonal are the squares of the values on the diagonal of the original matrix. For two matrices to be similar, they must have the same eigen values. In general A and A^2 will have different eigen values. In some rare cases they might be similar.