We present results from a minisurvey of relatively high redshift (1.7 < z < 4), gravitationally lensed, radio-quiet quasars observed with the Chandra X-Ray Observatory and with XMM-Newton. The lensing magnification effect allows us to search for changes with redshift in quasar spectroscopic and flux variability properties, ranging over 3 orders of magnitude in intrinsic X-ray luminosity. It extends the study of quasar properties to unlensed X-ray flux levels as low as a few times 10(-15) ergs cm(-2) s(-1) in the observed 0.4 - 8 keV band. For the first time, these observations of lensed quasars have provided medium to high signal-to-noise ratio X-ray spectra of a sample of relatively high redshift and low X-ray luminosity quasars. We find a possible correlation between the X-ray power-law photon index and the X-ray luminosity of the gravitationally lensed, radio-quiet quasar sample. The X-ray spectral slope steepens as the X-ray luminosity increases. This correlation is still significant when we combine our data with other samples of radio-quiet quasars with z > 1.5, especially in the low-luminosity range between 10(43) and 10(45.5) ergs s(-1). This result is surprising, considering that such a correlation is not found for quasars with redshifts below 1.5. We suggest that this correlation can be understood in the context of the hot-corona model for X-ray emission from quasar accretion disks, under the hypothesis that the quasars in our sample accrete very close to their Eddington limits and that the observed luminosity range is set by the range of black hole masses ( this hypothesis is consistent with recent predictions of semianalytic models for quasar evolution). The upper limits of X-ray variability for our relatively high redshift sample of lensed quasars are consistent with the known correlation between variability and luminosity observed in Seyfert 1 galaxies when this correlation is extrapolated to the larger luminosities of our sample.