  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 Keck/NIRC2 and VLT/NACO were obtained over a time
baseline of 5.8 years, delivering an astrometric accuracy of 0.5-1 mas/yr.
Proper motions were derived in the cluster reference frame and were used to
distinguish cluster members from the majority of field stars. Fitting the
cluster and field proper motion distributions with 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 with respect to the field along
the Galactic plane, which yields a 3D orbital velocity of 167 +/- 15 km/s 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
in the core of the Quintuplet cluster of sigma_1D < 10 km/s. Knowledge of the
three velocity components of the Quintuplet allows us to model the cluster
orbit in the potential of the inner Galaxy. Comparing the Quintuplet's orbit
with the Arches orbit, we discuss the possibility that both clusters originated
in the same area of the central molecular zone. [abridged]
