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This seems to work fine in Sage 8.4, for instance as tried on SageMathCell at https://sagecell.sagemath.org/.
Here is how to check the version of Sage (a good thing to include when reporting errors).
sage: print(version())
SageMath version 8.4, Release Date: 2018-10-17
Testing if two conjugacy classes are equal works:
sage: S = SymmetricGroup(3)
sage: a = Permutation('(1, 2, 3)')
sage: b = Permutation('(1, 3, 2)')
sage: c = Permutation('(1, 2)')
sage: d = Permutation('(1, 3)')
sage: G = PermutationGroup([a, c])
sage: H = PermutationGroup([b, d])
sage: GG = ConjugacyClass(S, G)
sage: HH = ConjugacyClass(S, H)
sage: GG == HH
False
I changed the notation from that in the question, but copy-pasting the exact code from the question works equally well (try it in SageMathCell).
