The problem of quadratic filtering for a bilinear stochastic system with state-dependent multiplicative noise and single delayed measurements was studied. Due to the presence of multiplicative noises, the system parameter matrices were random. So, the classical Kalman approach could not be directly used in the presence of multiplicative noises and would delay the measurements. Based on Kronecker algebra, the original system was changed into a linear augmented system, whose states and observations included the original states, observations and their second order Kronecker product. Then, the augmented system was transformed into a delay free system via the reorganized innovation approach, and the linear optimal filter for the augmented system was designed through projection theorem. Finally, the quadratic filter of the original system was derived by extracting the first n elements of the augmented state estimation. Compared with the widely used linear optimal filter, estimation accuracy of the quadratic filter increased 27％, and the overall performance was improved.