The elemental abundances in the broad-line regions of high-redshift quasars trace the chemical evolution in the nuclear regions of massive galaxies in the early Universe. In this work, we study metallicity-sensitive broad emission-line flux ratios in rest-frame UV spectra of 25 high-redshift (5.8 < z < 7.5) quasars observed with the VLT/X-shooter and Gemini/GNIRS instruments, ranging over log (M-BH/M-circle dot) = 8.4 - 9.8 in black hole mass and log (L-bol/erg s(-1)) = 46.7 - 47.7 in bolometric luminosity. We fit individual spectra and composites generated by binning across quasar properties: bolometric luminosity, black hole mass, and blueshift of the C IV line, finding no redshift evolution in the emission-line ratios by comparing our high-redshift quasars to lower redshift (2.0 < z < 5.0) results presented in the literature. Using CLOUDY-based locally optimally emitting cloud photoionization model relations between metallicity and emission-line flux ratios, we find the observable properties of the broad emission lines to be consistent with emission from gas clouds with metallicity that are at least 2-4 times solar. Our high-redshift measurements also confirm that the blueshift of the C IV emission line is correlated with its equivalent width, which influences line ratios normalized against C IV. When accounting for the C IV blueshift, we find that the rest-frame UV emission-line flux ratios do not correlate appreciably with the black hole mass or bolometric luminosity.