We look for signs of the H~I transverse proximity effect in the spectra of 130 QSO pairs, most with transverse separations in the plane of the sky of 0.1 -- 3 Mpc at z ~ 2.2. We expected to see a decrease in Lyman-alpha forest HI absorption in the spectrum of background QSOs near the position of foreground QSOs. Instead we see no change in the absorption in front of the foreground QSOs, and we see evidence for a 50% increase in the absorption out to 6 Mpc behind the foreground QSOs. Further, we see no change in the H I absorption along the line-of-sight to the foreground QSOs, the normal line-of-sight proximity effect. We may account for the lack of change in the H I absorption if the effect of extra UV photons is canceled by higher gas density around QSOs. If so, the increase in absorption behind the QSOs then suggests that the higher gas density there is not canceled by the UV radiation from the QSOs. We can explain our observations if QSOs have had their current UV luminosities for less than approximately a million years, a time scale that has been suggested for accretion disk instabilities and gas depletion.