The description of gene interactions that constantly occur in the cellular environment is an extremely challenging task due to an immense number of degrees of freedom and incomplete knowledge about microscopic details. Hence, a coarse-grained and rather powerful modeling of such dynamics is provided by Boolean Networks (BNs). BNs are dynamical systems composed of Boolean agents and a record of their possible interactions over time. Stable states in these systems are called attractors which are closely related to the cellular expression of biological phenotypes. Identifying the full set of attractors is, therefore, of substantial biological interest. However, for conventional high-performance computing, this problem is plagued by an exponential growth of the dynamic state space. Here, we demonstrate a novel quantum search algorithm inspired by Grover's algorithm to be implemented on quantum computing platforms. The algorithm performs an iterative suppression of states belonging to basins of previously discovered attractors from a uniform superposition, thus increasing the amplitudes of states in basins of yet unknown attractors. This approach guarantees that a new attractor state is measured with each iteration of the algorithm, an optimization not currently achieved by any other algorithm in the literature. Tests of its resistance to noise have also shown promising performance on devices from the current Noise Intermediate Scale Quantum Computing (NISQ) era.