In this paper, we introduce exploitation to the Beverton-Holt equation in the quantum calculus time setting. We first give a brief introduction to quantum calculus and to the Beverton-Holt q-difference equation before formulating the harvested Beverton-Holt q-difference equation. Under the assumption of a periodic carrying capacity and periodic inherent growth rate, we derive its unique periodic solution, which globally attracts all solutions. We further derive the optimal harvest effort for the Beverton-Holt q-difference equation under the catch-per-effort hypothesis. Examples are provided and discussed in the last section.