This program is part of QUEST (Quasar/ULIRG Evolutionary Study) and seeks to examine the gaseous environments of z < 0.3 quasars and ULIRGs as a function of host galaxy properties and age across the merger sequence from ULIRGs to quasars. This first paper in the series focuses on 33 quasars from the QUEST sample and on the kinematics of the highly ionized gas phase traced by the N V 1238, 1243 A and O VI 1032, 1038 A absorption lines in high-quality Hubble Space Telescope (HST) Cosmic Origins Spectrograph (COS) data. N V and O VI outflows are present in about 60% of the QUEST quasars and span a broad range of properties, both in terms of equivalent widths (from 20 mA to 25 A) and kinematics (outflow velocities from a few x 100 km/s up to ~10,000 km/s). The rate of incidence and equivalent widths of the highly ionized outflows are higher among X-ray weak or absorbed sources. The weighted outflow velocity dispersions are highest among the X-ray weakest sources. No significant trends are found between the weighted outflow velocities and the properties of the quasars and host galaxies although this may be due to the limited dynamic range of properties of the current sample. These results will be re-examined in an upcoming paper where the sample is expanded to include the QUEST ULIRGs. Finally, a lower limit of ~0.1% on the ratio of time-averaged kinetic power to bolometric luminosity is estimated in the 2-4 objects with blueshifted P V 1117, 1128 absorption features.