Based on time-series observations collected from the Zwicky Transient Facility (ZTF), we derived period-luminosity-metallicity (PLZ) and period-Wesenheit-metallicity (PWZ) relations for RR Lyrae located in globular clusters. We have applied various selection criteria to exclude RR Lyrae with problematic or spurious light curves. These selection criteria utilized information on the number of data points per light curve, amplitudes, colors, and residuals on the period-luminosity and/or period-Wesenheit relations. Due to blending, a number of RR Lyrae in globular clusters were found to be anomalously bright and have small amplitudes of their ZTF light curves. We used our final sample of ~750 RR Lyrae in 46 globular clusters covering a wide metallicity range (-2.36dex<[Fe/H]<-0.54dex) to derive PLZ and PWZ relations in the gri bands. In addition, we have also derived the period-color-metallicity and, for the first time, the period-Q-index-metallicity relations, where the Q-index is extinction-free by construction. We have compared our various relations to empirical and theoretical relations available in the literature and found a good agreement with most studies. Finally, we applied our derived PLZ relation to a dwarf galaxy, Crater II, and found that its true distance modulus should be larger than the most recent determination.