Herein, UiO-66 and its two functional analogs (with -NO2 and -NH2 functional groups) were synthesized, and their potential ability as pH stimulus nanocarriers of quercetin (QU), an anticancer agent, was studied. UiO-66 is a low-toxicity, biocompatible metal-organic framework with a large surface area and good stability, which can be prepared through a facile and inexpensive method. Before and after drug loading, various analyses were conducted to characterize the synthesized nanocarriers. Moreover, Monte Carlo simulations were performed to investigate their structures and interactions with quercetin. The most promising drug loading potential and prolonged drug release (over 25 days) were observed in QU@UiO-66-NO2 with 37% drug loading content, which was the best-tested sample that exhibited a higher release rate under acidic conditions (pH = 5) than that in normal cells (pH = 7.4). This behavior is known as pH-stimulus-controlled ability. The cell treatment with free QU, UiO-66-R, and QU@UiO-66-R (R = -H, -NO2, and -NH2) was performed, and an MTT assay was conducted on HEK-293 and MDA-MB-231 cells for the cytotoxicity study. Additionally, the kinetic modeling of drug release was investigated on the basis of the analysis of the drug release profiles.