Quaternion adaptive filters have been widely used in processing three-dimensional and four-dimensional signals. To improve the performance of quaternion adaptive filtering algorithms, this letter first proposes a novel derivation rule named generalized HR (GHR) $q$-derivative based on the concept of $q$-derivative and quaternion GHR derivative. Then, the product rules of GHR $q$-derivative are deduced, and the results of GHR $q$-derivative regarding some common univariate and multivariable functions are given. In addition, based on the proposed GHR $q$-derivative, the cost function of the quaternion least mean square (QLMS) algorithm is minimized to generate the $q$-QLMS algorithm. Finally, an example of Lorentz chaotic time-series prediction verifies the effectiveness of the proposed $q$-QLMS algorithm.