Preserving the underlying statistical properties of light during frequency transduction is essential for interfacing spectrally disparate quantum systems. To do this successfully, the process needs to be highly efficient, broadband and noiseless. Despite significant progress in this field, satisfying all these conditions remains challenging. Recently, molecular modulation-based transduction in hollow-core fibers has emerged as a promising approach, where biphoton correlations were proven to be preserved during frequency up-conversion. However, beyond this previous demonstration of correlation-maintaining conversion, whether molecular modulation can preserve more general statistical properties of light remains experimentally unexplored. Here, we demonstrate the conservation of photon-number statistics during selective frequency transduction by Raman molecular modulation in a hydrogen-filled hollow-core fiber. As a proof of concept, weak 585-nm signals are selectively converted to 773 nm in the near-infrared and to 471 nm in the blue spectral region with near-unity efficiencies. Phase matching between the generated coherence waves and the transitions of interest is achieved through the effective dispersion of the system, enabling independent selection of the frequency up- and down-conversion processes. The preservation of the photon-number statistics of the shifted signals, along with the large frequency shifts, broad tunability and compatibility with quantum-optical fields, provides a versatile fiber-based platform for quantum frequency transduction. In particular, the demonstrated interfacing between visible and near-infrared wavelengths using low pump energies and low gas pressures offers a route towards connecting spectrally distinct quantum systems, including quantum emitters, sensors and atomic quantum memories, within hybrid quantum networks.