In this paper, we study the Beverton-Holt equation with periodic inherent growth rate and periodic carrying capacity in the quantum calculus time setting. After a brief introduction to quantum calculus, we solve the Beverton-Holt q-difference equation using the logistic transformation. This leads to a linear q-difference equation where the solution is obtained using variation of parameters. The analysis of the solution aids our investigation of the first and second Cushing-Henson conjectures under the assumption of a periodic growth rate and a periodic carrying capacity. The first Cushing-Henson conjecture holds in the classical sense, which guarantees the existence of a unique periodic solution which is globally attractive. The analysis of the average of the unique periodic solution of the Beverton-Holt q-difference equation yields formulations of modified second Cushing-Henson conjectures.