Abstract: This paper is concerned with a linear-quadratic optimal control problem for backward stochastic differential equations driven by a Brownian motion and a Poisson random measure in an infinite dimensional Hilbert space. We first systematically formulate the definition of the solution of an infinite dimensional backward stochastic differential equation and the Yosida approximation technique introduced to overcome the limitations of the applicability of the Ito.s formula in the infinite dimensional case. Second, based on the convex variation principle, the existence and uniqueness of the solution to the optimal control problem is proved. Based on this, by constructing the dual equation of the state equation, we derive the stability conditions of the optimal control problem and construct a stochastic Hamiltonian system for infinite dimensional systems. Finally, we obtain an explicit expression for the optimal control by the solution of the Riccati equation.