The first passage time (FPT) is a generic measure that quantifies when a random quantity reaches a specific state. We consider the FTP distribution in nonlinear stochastic biochemical networks, where obtaining exact solutions of the distribution is a challenging problem. Even simple two-particle collisions cause strong nonlinearities that hinder the theoretical determination of the full FPT distribution. Previous research has either focused on analyzing the mean FPT, which provides limited information about a system, or has considered time-consuming stochastic simulations that do not clearly expose causal relationships between parameters and the system's dynamics. This paper presents the first exact theoretical solution of the full FPT distribution in a broad class of chemical reaction networks involving $A + B \rightarrow C$ type of second-order reactions. Our exact theoretical method outperforms stochastic simulations, in terms of computational efficiency, and deviates from approximate analytical solutions. Given the prevalence of bimolecular reactions in biochemical systems, our approach has the potential to enhance the understanding of real-world biochemical processes.