In many applications of state estimation, the process noise is colored; this case is addressed by applying the standard Kalman filter (KF) to dynamics that are augmented with the coloring dynamics. The present paper considers the case where the coloring dynamics are unknown, which renders the estimates obtained from the standard approach suboptimal. To address this problem, the present paper proposes an adaptive technique based on the principle that, if the measurement noise is white, then the innovations sequence is white if and only if the process noise is white. Leveraging this fact, an Innovations-Whitening Adaptive Kalman Filter (IWAKF) is developed, which learns the process-noise coloring online. By embedding an unknown coloring filter in a state-augmentation framework, IWAKF adapts its parameters by minimizing the empirical autocorrelation of the innovations, thereby driving them toward whiteness and restoring near-optimality without prior knowledge of the coloring dynamics.