The development of dual-stimuli-responsive hydrogels attracts much research interest owing to its unique stimuli-responsive characteristics. In this study, a poly-N-isopropyl acrylamide-co-glycidyl methacrylate-based copolymer was synthesized by incorporating N-isopropyl acrylamide (NIPAm) and a glycidyl methacrylate (GMA) monomer. The synthesized copolymer, pNIPAm-co-GMA was further modified with L-lysine (Lys) functional units and further conjugated with fluorescent isothiocyanate (FITC) to produce a fluorescent copolymer pNIPAAm-co-GMA-Lys hydrogel (HG). The in vitro drug loading and dual pH- and temperature-stimuli-responsive drug release behavior of the pNIPAAm-co-GMA-Lys HG was investigated at different pH (pH 7.4, 6.2, and 4.0) and temperature (25 °C, 37 °C, and 45 °C) conditions, respectively, using curcumin (Cur) as a model anticancer drug. The Cur drug-loaded pNIPAAm-co-GMA-Lys/Cur HG showed a relatively slow drug release behavior at a physiological pH (pH 7.4) and low temperature (25 °C) condition, whereas enhanced drug release was achieved at acidic pH (pH 6.2 and 4.0) and higher temperature (37 °C and 45 °C) conditions. Furthermore, the in vitro biocompatibility and intracellular fluorescence imaging were examined using the MDA-MB-231 cell line. Therefore, we demonstrate that the synthesized pNIPAAm-co-GMA-Lys HG system with temperature- and pH-stimuli-responsive features could be promising for various applications in biomedical fields, including drug delivery, gene delivery, tissue engineering, diagnosis, antibacterial/antifouling material, and implantable devices.