This paper studies the tracking problem of an uncertain LTI system where both the control input and system state are quantized. The L1 adaptive controller is designed for the quantized system. Two common types of quantization, logarithmic and uniform, quantization are considered. In both cases, the analysis of the closed-loop system provides a uniform transient performance bound, which depends on the adaptation rate and the quantization densities of the state and the input. By increasing the adaptation rate and improving the state and the input quantization, the closed-loop system response can be rendered arbitrarily close to the reference system output. Finally, the simulations illustrate the theoretical results.