A Dynkin game is considered for stochastic differential equations with random coefficients. We first apply Qiu and Tang's maximum principle for backward stochastic partial differential equations to generalize a Krylov estimate for the distribution of a Markov process to that of a non-Markov process and establish a generalized Ito-Kunita-Wentzell formula which allows the test function as an Ito process to take values in a Sobolev space. We then prove the verification theorem that the Nash equilibrium point and the value of the Dynkin game are characterized by the strong solution of the associated Hamilton-Jacobi-Bellman-Isaacs equation, which is currently a backward stochastic partial differential variational inequality (BSPDVI) with two obstacles. Finally, we prove the existence and uniqueness result and a comparison theorem for strong solution of the BSPDVI.