How to find an example of a 6 by 6 symmetric matrix (which is not identity matrix and does not satisfy $A^2=\text{ Identity matrix}$) which is similar to its inverse?

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Let me describe three constructions of matrices which are similar to their inverses over Z. It has little to do with SageMath. That is rather linear algebra.

Construction 1: There are infinitely 2x2 matrices with integral coefficients that are similar over Z to their inverses

sage: A = matrix(2, [1,1,0,1])
sage: B = matrix(2, [1,0,1,1])
sage: P = matrix(2, [-1,0,0,1])
sage: C1 = A
sage: C2 = B
sage: C3 = A * B * A
sage: all(P * C * P**-1 == C**-1 for C in [C1, C2, C3])
True

You can just build your 6x6 matrix M as a block matrix

sage: M = matrix(ZZ, 6)
sage: M[0:2,0:2] = C1
sage: M[2:4,2:4] = C2
sage: M[4:6,4:6] = C3
sage: Q = matrix(ZZ, 6)
sage: Q[0:2,0:2] = Q[2:4,2:4] = Q[4:6,4:6] = P
sage: Q * M * Q**-1 == M**-1
True

Construction 2: The companion matrix of a symmetric polynomial

sage: x = polygen(ZZ, 'x')
sage: a = 13
sage: b = 25
sage: c = 18
sage: M = companion_matrix(X**6 + a*X**5 + b*X**4 + c*X**3 + b*X**2 + a*X + 1)
sage: S = SymmetricGroup(6)
sage: P = S('(1,6)(2,5)(3,4)').matrix()
sage: P * M * P**-1 == M**-1
True

Construction 3: Permutation matrices

sage: S = SymmetricGroup(6)
sage: M = S('(1,2,3,4)(5,6)').matrix()
sage: P = S('(1,4)(2,3)').matrix()
sage: P * M * P**-1 == M**-1
True