Brillouin imaging (BI) for micromechanical characterization of tissues and biomaterials is a fast-developing field of research with a strong potential for medical diagnosis of disease-modified tissues and cells. Although the principles of BI imply its compatibility with in vivo and in situ measurements, the integration of BI with a flexible catheter, capable of reaching the region of interest within the body, is yet to be reported. Here, for the first time, we experimentally investigate integration of the Brillouin spectroscope with standard optical fiber components to achieve a Brillouin endoscope. The performance of single-fiber and dual-fiber endoscopes are demonstrated and analyzed. We show that a major challenge in construction of Brillouin endoscopes is the strong backward Brillouin scattering in the optical fiber and we present a dual-fiber geometry as a possible solution. Measurements of Brillouin spectra in test liquids (water, ethanol and glycerol) are demonstrated using the dual-fiber endoscope and its performance is analyzed numerically with the help of a beam propagation model.