Quantum networks (QNs) relying on free-space optical (FSO) quantum channels can support quantum applications in environments wherein establishing an optical fiber infrastructure is challenging and costly. However, FSO-based QNs require a clear line-of-sight (LoS) between users, which is challenging due to blockages and natural obstacles. In this paper, a reconfigurable intelligent surface (RIS)-assisted FSO-based QN is proposed as a cost-efficient framework providing a virtual LoS between users for entanglement distribution. A novel modeling of the quantum noise and losses experienced by quantum states over FSO channels defined by atmospheric losses, turbulence, and pointing errors is derived. Then, the joint optimization of entanglement distribution and RIS placement problem is formulated, under heterogeneous entanglement rate and fidelity constraints. This problem is solved using a simulated annealing metaheuristic algorithm. Simulation results show that the proposed framework effectively meets the minimum fidelity requirements of all users’ quantum applications. This is in stark contrast to baseline algorithms that lead to a drop of at least 84% in users’ end-to-end fidelities. The proposed framework also achieves a 63% enhancement in the fairness level between users compared to baseline rate maximizing frameworks. Finally, the weather conditions, e.g., rain, are observed to have a more significant effect than pointing errors and turbulence.