We found new two-dimensional (2D) quantum (S=1/2) antiferromagnetic systems: CuRE₂Ge₂O₈ (RE=Y and La). According to our analysis of high-resolution x-ray and neutron diffraction experiments, the Cu network of CuRE₂Ge₂O₈ (RE=Y and La) exhibits a 2D triangular lattice linked via weak bonds along the perpendicular b axis. Our bulk characterizations from 0.08 to 400 K show that they undergo a long-range order at 0.51(1) and 1.09(4) K for the Y and La systems, respectively. Interestingly, they also exhibit field induced phase transitions. For theoretical understanding, we carried out the density functional theory (DFT) band calculations to find that they are typical charge-transfer-type insulators with a gap of Eg≅2eV. Taken together, our observations make CuRE₂Ge₂O₈ (RE=Y and La) additional examples of low-dimensional quantum spin triangular antiferromagnets with the low-temperature magnetic ordering.