This paper deals with a particular case of Bi-Objective Optimization called \textit{Bi-Objective Discrete Optimization} (BODO), where the feasible set is discrete, and the two objectives take only positive values. Since the feasible set of a BODO problem is discrete and usually finite, it can theoretically be enumerated to identify the Pareto set, which consists of all Pareto-optimal solutions representing different trade-offs between two objectives. However, in general, this problem is challenging due to two main issues: time complexity, as the number of Pareto-optimal solutions can be exponentially large, and lack of decisiveness. From a practical point of view, the Central Decision Maker (CDM) may be interested in a reduced Pareto set reflecting the own preference of the CDM, which a computationally tractable algorithm can obtain. In this paper, we propose a new criterion for selecting solutions within the Pareto set of BODO. For this purpose, we focus on solutions achieving proportional fairness between two objectives, called \textit{generalized Nash Fairness solutions} ($\rho$-NF solutions). The positive parameter $\rho$ provided by the CDM reflects the relative importance of the first objective compared to the second one. We first introduce the $\rho$-NF solution concept for BODO. We then show that the $\rho$-NF solution set is a subset of the Pareto set, and this inclusion can be strict. We also propose a recursive Newton-like algorithm for determining the $\rho$-NF solution set. Finally, an illustrative example of BODO is given.