When we say that an operator is Hermitian in QM, does it depend on the Hilbert space under consideration, or not? Are there operators that are Hermitian in one Hilbert space but not in another?
There is a unique Hilbert space upto isomorphism for any given dimension.
So, a Hermetian operator on one Hilbert space of dimension $d$ is also a Hermetian operator on any other Hilbert space of the same dimension $d$.
So the answer to your question is no.