This paper considers a chemotaxis model coupled a stochastic Navier-Stokes equation in two-dimensional case. In the non-convex bounded domain, it is proved that the stochastic chemotaxis-Navier-Stokes system possesses at least one global martingale weak solution when the chemotactic sensitivity function $\chi$ is nonsingular. In the convex bounded domain, under the conditions of $\chi$ and per capita oxygen consumption rate $h$ are appropriately relaxed (where $\chi$ allows singularity), it is proved that the system admits a unique global mild solution. Our results generalize previously known ones.