  The relaxation of a quantum emitter (QE) near metal-dielectric layered
nanostructures is investigated, with focus on the influence of plasmonic
quantum effects. The Green's tensor approach, combined with the Feibelman
$d$-parameter formalism, is used to calculate the Purcell factor and the
dynamics of a two-level QE in the presence of the nanostructure. Focusing on
the case of Na, we identify electron spill-out as the dominant source of
quantum effects in jellium-like metals. Our results reveal a clear splitting in
the emission spectrum of the emitter, and non-Markovian relaxation dynamics,
implying strong light--matter coupling between them, a coupling that is not
prevented by the quantum-informed optical response of the metal.
