Because of the strong coupling between the magnetic and dielectric properties, the study of quantum paraelectric EuTiO3 has attracted more and more attention in both theoretical and experimental research recently. In this paper, the first principles based on the density functional theory within the generalized gradient approxiamtion is used to investigate the magnetic and electronic structure of quantum paraelectric EuTiO3, and to analyze the effects of the strain on the magnetic and strutural phase transition, in turn to discuss the possible magnetoelectric coupling mechanism of this material. The calculations show that EuTiO3 with the strain-free is in a paraelectric cubic and G-type antiferromagnetic state at low temperature, while appling either compressive or tensile strain along the c-axis to it, the balance of hybridization between Ti 3d and O 2p orbit will be breaken and EuTiO3 will transite from paraelectric and G-antiferromagnetic to ferroelectric-ferromagnetic structure as the strain is increased to a certain value. All those indicate the strong spin-lattice coupling effect in EuTiO3.