The study of the properties for fractional stochastic differential equation is one of the hot directions in the field of mathematics over the years. For a class of linear fractional stochastic differential equation on Hilbert space ，the existence and uniqueness of its square-mean asymptotically almost periodic mild solutions are studied，and then the conclusions of this kind of linear fractional stochastic differential equation are extended to corresponding semi-linear fractional stochastic differential equation. The existence and uniqueness of square-mean asymptotically almost periodic mild solutions for this kind of semi-linear fractional stochastic differential equation are discussed by Banach fixed point theorem，and then discuss the existence of square-mean asymptotically almost periodic mild solutions by using Schauder fixed point theorem under non-Lipschitz conditions.