We report the spectroscopic confirmation of a merging pair of massive quiescent galaxies at z = 3.44. Using JWST observations, we confirm that the two galaxies lie at a projected separation of 4.5 kpc with a velocity offset of similar to 680 km s(-1) (delta(z) similar to 0.01). The pair resides in the core of a known rich overdensity of galaxies, dubbed the "Cosmic Vine". For both pair members, modeling of the spectral energy distributions and faint rest-frame optical emission lines indicate high stellar masses (log(M-star/M-circle dot) similar to 10.9) and suppressed star formation (log(sSFR/yr(-1)) < -10), more than an order of magnitude below the level of the star formation main sequence at this redshift. We then explore the Illustris-TNG simulation and the GAEA and SHARK semi-analytical models to examine whether they produce a pair of massive quiescent galaxies akin to that of the Cosmic Vine. While all models produce close pairs of massive quiescent galaxies at 2 < z < 4 with comparable separations and velocity offsets, their predicted number densities are 10-80 times lower than our observational constraint. This discrepancy cannot be fully explained by coarse time sampling in these models or the general challenge of forming early massive quiescent galaxies in simulations. Given that > 90% of simulated pairs in the models that we analyzed merge by z = 0, our findings suggest that our observed pair will likely coalesce into a single massive galaxy. The merger, occurring in the dense core of a large-scale structure, might represent a critical event in the formation of a brightest cluster galaxy and the morphological transformation of high-redshift disky quiescent galaxies into early-type ellipticals.