Two‐dimensional (2D) conjugated metal‐organic frameworks (c‐MOFs) are promising materials for supercapacitor (SC) electrodes due to their high electrochemically accessible surface area coupled with superior electrical conductivity compared to traditional MOFs. Here, porous and non‐porous HHB‐Cu (HHB=hexahydroxybenzene), derived through surfactant‐assisted synthesis, are studied as representative 2D c‐MOF models, showing different reversible redox reactions with Na+ and Li+ in aqueous and organic electrolytes, respectively. We deployed these redox activities to design negative electrodes for hybrid SCs (HSCs), combining the battery‐like property of HHB‐Cu as negative electrode and the high capacitance and robust cyclic stability of activated carbon (AC) as positive electrode. In organic electrolyte, porous HHB‐Cu‐based HSC achieves a maximum cell specific capacity (Cs) of 22.1 mAhg‐1 at 0.1 Ag‐1, specific energy (Es) of 15.55 Whkg‐1 at specific power (Ps) of 70.49 Wkg‐1, and 77% cyclic stability after 3000 gravimetric charge‐discharge (GCD) cycles at 1 Ag‐1 (calculated on the mass of both electrode materials). In the aqueous electrolyte, porous HHB‐Cu‐based HSC displays a Cs of 13.9 mAhg‐1 at 0.1 Ag‐1, Es of 6.13 Whkg‐1 at 44.05 Wkg‐1, and 72.3% Cs retention after 3000 GCD cycles. The non‐porous sample shows lower Es performance but better rate capability compared to the porous one.