We investigate the orbital motion of the Quintuplet cluster near the Galactic center with the aim of constraining formation scenarios of young, massive star clusters in nuclear environments. Three epochs of adaptive optics high-angular resolution imaging with the Keck/NIRC2 and Very Large Telescope/NAOS-CONICA systems were obtained over a time baseline of 5.8 yr, delivering an astrometric accuracy of 0.5-1 mas yr-1. Proper motions were derived in the cluster reference frame and were used to distinguish cluster members from the majority of the dense field star population toward the inner bulge. Fitting the cluster and field proper motion distributions with two-dimensional (2D) Gaussian models, we derive the orbital motion of the cluster for the first time. The Quintuplet is moving with a 2D velocity of 132 ± 15 km s-1 with respect to the field along the Galactic plane, which yields a three-dimensional orbital velocity of 167 ± 15 km s-1 when combined with the previously known radial velocity. From a sample of 119 stars measured in three epochs, we derive an upper limit to the velocity dispersion of σ1D < 10 km s-1 in the core of the Quintuplet cluster. Knowledge of the three velocity components of the Quintuplet allows us to model the cluster orbit in the potential of the inner Galaxy. Under the assumption that the Quintuplet is located in the central 200 pc at the present time, these simulations exclude the possibility that the cluster is moving on a circular orbit. Comparing the Quintuplet's orbit with our earlier measurements of the Arches' orbit, we discuss the possibility that both clusters originated in the same area of the central molecular zone (CMZ). According to the model of Binney et al., two families of stable cloud orbits are located along the major and minor axes of the Galactic bar, named x1 and x2 orbits, respectively. The formation locus of these clusters is consistent with the outermost x2 orbit and might hint at cloud collisions at the transition region between the x1 and x2 orbital families located at the tip of the minor axis of the Galactic bar. The formation of young, massive star clusters in circumnuclear rings is discussed in the framework of the channeling in of dense gas by the bar potential. We conclude that the existence of a large-scale bar plays a major role in supporting ongoing star and cluster formation, not only in nearby spiral galaxies with circumnuclear rings, but also in the Milky Way's CMZ. © 2014. The American Astronomical Society. All rights reserved.