About 10 percent of quasars are known to exhibit deep broad absorption troughs blueward of prominent permitted emission lines, which are usually attributed to the existence of outflows slightly above he accretion disk around the supermassive black hole. Typical widths up to 0.2c of these absorption roughs indicate the velocity scales in which special relativistic effects may not be negligible. Under he assumption of the ubiquity of the broad absorption line region in quasars, the broad emission line flux will exhibit Thomson scattered components from these fast outflows. In this paper, we provide our Monte Carlo calculation of linear polarization of singly Thomson scattered line radiation with the careful considerations of special relativistic effects. The scattering region is approximated by a collection of rings that are moving outward with speeds υ =cβ < 0.2c near the equatorial plane, and the scattered line photons are collected according to its direction and wavelength in the observer's rest frame. We find that the significantly extended red tail appears in the scattered radiation. We also find that the linear degree of polarization of singly Thomson scattered line radiation is wavelength-dependent and hat there are significant differences in the linear degree of polarization from that computed from classical physics in the far red tail. We propose that the semi-forbidden broad emission line C III]1909 may be significantly contributed from Thomson scattering because this line has small resonance scattering optical depth in the broad absorption line region, which leads to distinct and significant polarized flux in this broad emission line.