The light we observe from distant astrophysical objects including supernovae and quasars allows us to determine large distances in terms of a cosmological model. Despite the success of the standard cosmological model in fitting the data, there remains no underlying explanation for the accelerated expansion and dark matter. Furthermore, there is a current tension between early- and late-universe determinations of the Hubble constant. New techniques may offer the possibility of measuring out to larger distances, provide complementary information, or be able to side-step current limitations. After reviewing the fundamentals of standard cosmology and gravitational lensing, this thesis investigates a novel method of cosmography based on combining the techniques of strong gravitational lensing time delay measurements and quasar reverberation mapping. The motivation for this method was the possibility of avoiding lens modelling challenges, such as the mass-sheet degeneracy, typically associated with time delay cosmography. It suggested that differential time delays originating from spatially separated signals in the Broad Line Region of a quasar could be distinguished and measured from the spectroscopy of the images, and utilised to provide a ratio of cosmological distances independent of the lensing potential. An analytic description of the effect of the differential lensing on the emission line spectral flux for axisymmetric Broad Line Region geometries is given, with the inclined ring or disk, spherical shell, and double cone as examples. This critical examination shows that the proposed method is unable to recover cosmological information, as the observed time delay and inferred line-of-sight velocity do not uniquely map to the three-dimensional position within the quasar.