The strong growth in the number of connected mobile devices has imposed new challenges in efficiently exploiting the available networks resources. Code Domain Non-Orthogonal Multiple Access (NOMA) technique appears as a tremendous efficient solution. Each device uses its assigned code to simultaneously transmit its data along with the user identifier, without any resource reservation exchange, saving precious wireless resources. However, this requires a receiver capable of blindly detecting the active users, which is highly complex. Driven by the promising superposition property of quantum architecture, the goal of this paper is to adapt and apply the quantum Grover algorithm for Active User Detection (AUD) purpose in the context of NOMA, to alleviate the search complexity. This adapted Grover's algorithm is compared with the optimal classical Maximum Likelihood (ML) AUD receivers, as well as with the basic classical Conventional Correlation Receiver (CCR). A benchmark on the probability of AUD is assessed as a function of the Signal to Noise Ratio (SNR) of the received signal. We show that our adapted Grover's algorithm is very promising in high SNR regime.