Quantum aspects, such as electron tunneling between closely separated metallic nanoparticles, are crucial for understanding the plasmonic response of nanoscale systems. We explore quantum effects on the response of the conductively coupled metallic nanoparticle dimer. This is realized by stretching a nanorod, which leads to the formation of a narrowing atomic contact between the two nanorod ends. Based on first-principles time-dependent density-functional-theory calculations, we find a discontinuous evolution of the plasmonic response as the nanorod is stretched. This is especially pronounced for the intensity of the main charge-transfer plasmon mode. We show the correlation between the observed discontinuities and the discrete nature of the conduction channels supported by the formed atomic-sized junction.