In this article, the problem of the stochastically finite time stabilization for an uncertain single-input and single-output stochastic system in presence of input quantization is studied. The broad learning system (BLS) is first applied to identify the uncertain system with unknown dynamics. The problem of unmeasured states can be solved by establishing a novel BLS-based state observer. Combining the stochastically finite time theorem with Itô formula, a new finite time design method is proposed, which can reduce the difficulty in designing controllers by traditional methods. A stochastically finite time quantized control method is presented by utilizing a new finite time design Lemma 3 and quantized input decomposition technique. The developed control approach can guarantee that the closed-loop system is semi-global finite-time stable in probability, and the convergence performances are well in presence of actuator quantization. The simulation on a chemical reactor is utilized to verify the proposed scheme, which demonstrates the advantage of BLS, as well as the validity of our control method.