This paper is concerned with the first passage discounted payoff criterion for continuous-time N-person nonzerosum stochastic games in countable states and Borel action spaces, where the discount factor is nonconstant. For the game with unbounded reward functions and transition rates, the optimality is over the set of all randomized history-dependent policies. After showing the uniqueness of the solution to the Shapley equation, we give some suitable conditions imposed on the primitive data for the existence of Nash equilibria. Then, a financial system is introduced to illustrate the applications of the main result here.