Differential game is a theory that studies the decision-making process in which two or more players exert their control effects on a moving system described by differential equations to achieve their respective optimal goals. It has been widely concerned because of its interesting mathematical properties and application value in the economic field. A class of partially observed nonzero-sum differential games for backward stochastic systems with random jumps is considered, in which the game system involves the random jumps and each player possesses different observed equation. For this type of partially observed stochastic differential game problem, under the condition that the control domain is convex, the convex variation and duality techniques are used to establish the maximum principle of Nash equilibrium point. Under the appropriate convexity assumption, it is proved that the necessity optimal condition is also the sufficiency optimal condition, which is the verification theorem of Nash equilibrium point. Using the above maximum principle, a partially observed linear quadratic (LQ) game for backward stochastic systems with jumps is studied, and the unique optimal control of the LQ game problem is obtained, where the state equation and the adjoint equation constitute a class of forward backward stochastic differential equations with jumps. Since the LQ model is often used to describe many financial and economic phenomena, it is expected that the partially observed LQ game results for backward stochastic systems with jumps can be widely used in these fields.