We use time-dependent density functional theory (TDDFT) within the jellium model to study the impact of quantum-mechanical effects on the self-interaction Green's function that governs the electromagnetic interaction between quantum emitters and plasmonic metallic nanoantennas. A semiclassical model based on the Feibelman parameters, which incorporates quantum surface-response corrections into an otherwise classical description, confirms surfaceenabled Landau damping and the spill out of the induced charges as the dominant quantum mechanisms strongly affecting the nanoantenna-emitter interaction. These quantum effects produce a redshift and broadening of plasmonic resonances not present in classical theories that consider a local dielectric response of the metals. We show that the Feibelman approach correctly reproduces the nonlocal surface response obtained by full quantum TDDFT calculations for most nanoantenna-emitter configurations. However, when the emitter is located in very close proximity to the nanoantenna surface, we show that the standard Feibelman approach fails, requiring an implementation that explicitly accounts for the nonlocality of the surface response in the direction parallel to the surface. Our study thus provides a fundamental description of the electromagnetic coupling between plasmonic nanoantennas and quantum emitters at the nanoscale.