This paper investigates the problem of asynchronous dynamic quantization for nonlinear systems with unknown initial conditions and subject to external disturbances and delays. We propose a novel two-level dynamic quantizer that only depends on the sign of the quantization error to quantize the actual input signal. Building on the sliding mode approach, the proposed quantizer is enabled to capture the state/input in a finite time and maintains the quantization error bounded after capturing. Sufficient conditions are provided to ensure closed-loop stability in terms of Lyapunov functions. Furthermore, the robustness of the quantization policy with respect to external disturbances and time delays is also investigated. The effectiveness of the approach is demonstrated through a numerical example.