Broad-line regions (BLRs) in high-redshift quasars provide crucial information on chemical enrichment in the early universe. Here we present a study of BLR metallicities in 33 quasars at redshift 5.7 < z < 6.4. Using the near-IR spectra of the quasars obtained from the Gemini telescope, we measure their rest-frame UV emission-line flux and calculate flux ratios. We then estimate BLR metallicities with empirical calibrations based on photoionization models. The inferred median metallicity of our sample is a few times the solar value, indicating that the BLR gas had been highly metal enriched at z ∼6. We compare our sample with a low-redshift quasar sample with similar luminosities and find no evidence of redshift evolution in quasar BLR metallicities. This is consistent with previous studies. The Fe ii/Mg ii flux ratio, a proxy for the Fe/α element abundance ratio, shows no redshift evolution as well, further supporting rapid nuclear star formation at z ∼6. We also find that the black hole mass-BLR metallicity relation at z ∼6 is consistent with the relation measured at 2 < z < 5, suggesting that our results are not biased by a selection effect due to this relation.