A novel generative site-specific beamforming (GenSSBF) framework is proposed, which integrates a site-information-maximizing (SIM) codebook with a conditional flow matching (CFM)-based beam generator. By this framework, the site-specific radio propagation environment is learned at the base station (BS), enabling the generation of high fidelity communication beams from coarse reference-signal-received-power (RSRP) feedback provided by user equipments (UEs). In the proposed design, a low-dimensional SIM probing codebook is first constructed by maximizing the mutual information between the RSRP feedback and the site-specific channel. This design not only reduces the initial beam sweeping overhead, but also enhances the amount of channel state information conveyed through UE feedback. By treating the RSRP feedback as a conditional prior, a CFM-based generative model is further developed to explicitly capture the uncertainty in beam generation. Specifically, a small set of UE-specific candidate beams is generated by inferring the learned generative model and sampling from the corresponding posterior distribution, after which the final data transmission beam is selected by the UE. Extensive simulation results demonstrate the effectiveness of both the proposed SIM codebook and the CFM-based beam generator. The proposed GenSSBF framework achieves beamforming performance nearly identical to maximum ratio transmission while requiring only eight probing beams and eight candidate beams.