The Schrödinger-Poisson-Landau-Lifshitz-Gilbert (SPLLG) system can characterize the spin transfer torque mechanism transferring the spin angular momentum to the magnetization dynamics through spin-magnetization coupling. We study the three-dimensional stochastic SPLLG system driven by a multiplicative stochastic force containing a continuous noise and a small jump noise. We establish the existence of weak martingale solutions based on the penalized functional technique, the Faedo-Galerkin approximation, stochastic compactness method, and a careful identification of the limit. Due to the strong coupling and strong nonlinearity caused by the SPLLG system and stochastic effects, some crucial difficulties have been encountered in obtaining energy estimates and avoiding non-negativity of the test function. We mainly utilize the structure of equations and the property of martingales developing the new energy estimates, and apply the three-layer approximation to overcome these difficulties. In particular, we extend the results by Z. Brzeźniak and U. Manna (Comm. Math. Phys., 2019) and by L.H. Chai, C.J. García-Cervera and X. Yang (Arch. Ration. Mech. Anal., 2018) to both the stochastic case and the coupling case.