This paper studies the identification of ARMA systems with colored measurement noises using finite-level quantized observations. Compared with the case under colorless noises, this problem is more challenging. Our approach is to jointly design an adaptive quantizer and a recursive estimator to identify system parameters. Specifically, the quantizer uses the latest estimate to adjust its thresholds, and the estimator is updated by using quantized observations. To accommodate the temporal correlations of quantization errors and measurement noises, we construct a second-order statistics equivalent system, from which the original ARMA system is identified. The associated identifiability problem and convergence are analyzed as well. Finally, numerical simulations are performed to demonstrate the effectiveness of the proposed algorithm.