Magnetic ground state of perovskite structure quantum paraelectric EuTiO3 has been known to have a planar anisotropic G-type antiferromagnet structure according to the experimental study. In this paper, based on density functional theory, first-principles computations are performed to investigate the magnetic properties and spin-exchange interaction of EuTiO3 in both of the quantum paraelectric phase and ferroelectric tetragonal phase under stress. By analyzing the energies of different magnetic structures and paths of spin exchange coupling as well as the effect of stress on the magnetic exchange paths change, it is found that when the system of EuTiO3 is free, it has a G-type antiferromagnetic structure with uniaxial anisotropic spin along [110] direction. Furthermore, in this structure, Eu 4f electron spin achieves antiferromagnetic super-exchange coupling via O 2p state at face-centered position. However, in the ferroelectric tetragonal phase structure induced by applied stress field, Eu 4f electron spin achieves ferromagnetic exchange coupling in [110] direction due to the variation of Eu-O-Eu bond angle in spin exchange path.