We present the results of an exploratory study of broad-line region (BLR) metallicity in 34 2.2 <= z <= 4.6 quasars with far-infrared (FIR) luminosities (L-FIR) from 10(13.4) to <= 10(12.1) L-circle dot. Quasar samples sorted by L-FIR might represent an evolutionary sequence if the star formation rates (SFRs) in quasar hosts generally diminish across quasar lifetimes. We use rest-frame ultraviolet spectra from the Sloan Digital Sky Survey to construct three composite spectra sorted by L-FIR, corresponding to average SFRs of 4980, 2130 and <= 340 M-circle dot yr(-1) after correcting for a nominal quasar FIR contribution. The measured N V lambda 1240/C IV lambda 1550 and Si IV lambda 1397+ O IV] lambda 1402/C IV lambda 1550 emission line ratios indicate supersolar BLR metallicities in all three composites, with no evidence for a trend with the star formation rate. The formal derived metallicities, Z similar to 5-9 Z(circle dot), are similar to those derived for the BLRs of other quasars at similar redshifts and luminosities. These results suggest that the ongoing star formation in the host is not responsible for the metal enrichment of the BLR gas. Instead, the BLR gas must have been enriched before the visible quasar phase. These results for high quasar metallicities, regardless of L-FIR, are consistent with evolution scenarios wherein visibly bright quasars appear after the main episode(s) of star formation and metal enrichment in the host galaxies. Finally, young quasars, those more closely associated with a recent merger or a blowout of gas and dust, may exhibit tracers of these events, such as redder continuum slopes and higher incidence of narrow absorption lines. With the caveat of small sample sizes, we find no relation between L-FIR and the reddening or the incidence of absorption lines.