If $\kappa$ is such that $2^\kappa > 2^{\aleph_0}$ then there are no $\kappa$-sized Q sets. In particular, no set of reals of size continuum is Q. This is because for any set of reals viewed as a topological space with the subspace topology, there are only continuum many Borel sets so if your space has more than continuum many subsets it cannot be Q. Therefore in fact ZFC proves there is a non-Q set without the perfect set property, without any hypothesis on the $\omega_1$ of L. See this paper for more info: https://arxiv.org/pdf/1611.08152.pdf