Recent observations of high-redshift galaxies and quasars indicate that the hydrogen component of the inter-galactic medium (IGM) must have been reionized at some redshift z greater than or similar to 6. Prior to complete reionization, sources of ultraviolet radiation will be seen behind intervening gas that is still neutral, and their spectra should show the red damping wing of the Gunn-Peterson trough. While this characteristic feature may, in principle, totally suppress the Ly alpha emission line in the spectra of the first generation of objects in the universe, we show here that the IGM in the vicinity of luminous quasars will be highly photoionized on several megaparsec scales as a result of the source emission of Lyman continuum photons. If the quasar lifetime is shorter than the expansion and gas recombination timescales, the volume ionized will be proportional to the total number of photons produced above 13.6 eV; the effect of this local photoionization is to greatly reduce the scattering opacity between the redshift of the quasar and the boundary of its H II region. We find that the transmission on the red side of the Ly alpha resonance is always greater than 50% for sources radiating a total of greater than or similar to 10(69.5) ionizing photons into the IGM. The detection of a strong Ly alpha emission line in the spectra of bright quasistellar objects shining for greater than or similar to 10(7) yr cannot then be used, by itself, as a constraint on the reionization epoch. The first signs of an object radiating prior to the transition from a neutral to an ionized universe may be best searched for in the spectra of luminous sources with a small escape fraction of Lyman continuum photons into the IGM or sources with a short duty cycle.