Time-periodic perturbations due to classical electromagnetic fields are useful to engineer the topological properties of matter using the Floquet theory. Here we investigate the effect of quantized electromagnetic fields by focusing on the quantized light-matter interaction on the edge state of a quantum spin Hall insulator. A Dicke-type superradiant phase transition occurs at arbitrary weak coupling, the electronic spectrum acquires a finite gap, and the resulting ground-state manifold is topological with a Chern number of ±1. When the total number of excitations is conserved, a photocurrent is generated along the edge, being pseudoquantized as ωln(1/ω) in the low-frequency limit and decaying as 1/ω for high frequencies with ω the photon frequency. The photon spectral function exhibits a clean Goldstone mode, a Higgs-like collective mode at the optical gap and the polariton continuum.