We assemble 3524 quasars from the Sloan Digital Sky Survey (SDSS) with repeated observations to search for variations of the narrow C IV {lambda}{lambda}1548, 1551 and Mg II {lambda}{lambda}2796, 2803 absorption doublets in spectral regions shortward of 7000{AA} in the observed frame, which corresponds to time-scales of about 150-2643d in the quasar rest frame. In these quasar spectra, we detect 3580 CIV absorption systems with z_abs_=1.5188-3.5212 and 1809 MgII absorption systems with z_abs_=0.3948-1.7167. In term of the absorber velocity ({beta}) distribution in the quasar rest frame, we find a substantial number of CIV absorbers with {beta}<0.06, which might be connected to absorption of quasar outflows. The outflow absorption peaks at {upsilon}~2000km/s and drops rapidly below this peak value. Among 3580 CIV absorption systems, 52 systems (~1.5 percent) show obvious variations in equivalent widths in the absorber rest frame (W_r_): 16 enhanced, 16 emerged, 12 weakened and 8 disappeared systems, respectively. We find that changes in W_r_{lambda}1548 are related neither to the time-scales of the two SDSS observations nor to absorber velocities in the quasar rest frame. Variable absorption in low-ionization species is important to constrain the physical conditions of the absorbing gas. There are two variable MgII absorption systems measured from SDSS spectra detected by Hacker et al. (2013, J/MNRAS/434/163). However, in our MgII absorption sample, we find that neither shows variable absorption with confident levels of >4{sigma} for {lambda}2796 lines and >3{sigma} for {lambda}2803 lines.