Magnetic properties of ZnO with intrinsic point defects, such as Co impurity, were calculated in the framework of quantum-mechanical Heisenberg model. Direct exchange in the chain model of magnetic ion impurities was studied, the magnetic chain was being created in (001) and (100) planes. In both cases the ground states of systems have antiferromagnetic order. Exchange interaction integrals were obtained (-1.5 and 0.5 meV). We have also calculated microscopic magnetic parameters: Curie temperature (Tc_in = 34.04 and Tc_out=10.7 K), stiffness constant (Din=82.4, Dout=25.8 meV Å2), saturation magnetization (Msat_in=2.88×105, Msat_out=2.89×105A/m), exchange constant (Ain=4.46× 10-13 and Aout=1.42×10-12 J/m), and Bloch constant (Bin=6.53×10-5Bout= 3.7×10-4 K3/2). For the energy of magnetic anisotropy of ZnO:Co structure it was found that the easy magnetization axis lies in (100) plane. The value of magnetic anisotropy for the whole cell is 0.06 meV. The anisotropy energy density is 5.54 × 104 J / m3.