In this work, a new class of stochastic gradient algorithm is developed based on q-calculus. Unlike the existing q-LMS algorithm, the proposed approach fully utilizes the concept of q-calculus by incorporating a time-varying q parameter. The proposed enhanced q-LMS (Eq-LMS) algorithm utilizes a novel, parameterless concept of error-correlation energy and normalization of signal to ensure high convergence, stability and low steady-state error. The proposed algorithm automatically adapts the learning rate with respect to the error. For evaluation purposes the system identification problem is considered. The necessary condition of convergence for the proposed algorithm is analyzed, and the validation of analytical findings and simulation results is discussed. Extensive experiments show better performance of the proposed Eq-LMS algorithm compared to the standard q-LMS approach.