In this letter, the performance of a quantum key distribution (QKD) free-space optical (FSO) system is analyzed while taking a generalized pointing error model into account. More specifically, closed-form expressions for the average received powers at both the legitimate receiver and eavesdropper are derived. In addition, their corresponding asymptotic results valid in the high telescope gain regime are also presented. To capture the secure performance, we also investigate the ratio of received powers at the eavesdropper and at the legitimate receiver. Further, in some special cases, we find the optimal telescope gains for the received powers at both the legitimate receiver and eavesdropper, as well as the power ratio, which is important and useful for a secure QKD FSO system design. Finally, some selected numerical results are presented to illustrate the mathematical formalism and validate the accuracy of the derived analytical expressions.