The optical spectra of objects classified as QSOs in the Sloan Digital Sky Survey Data Release 6 are analyzed with the aim of determining the value of the fine structure constant in the past and then checked for possible changes in the constant over cosmological timescales. The analysis is done by measuring the position of the fine structure lines of the [O III] doublet (λλ4959 and λλ5008) in QSO nebular emission. From the sample of QSOs at redshifts z < 0.8, a sub sample was selected on the basis of the amplitude and width of the [O III] lines. Two different methods were used to determine the position of the lines of the [O III] doublet, both giving similar results. Using a clean sample containing 1568 of such spectra, a value of Δα/α = (+2.4 ± 2.5) × 10-5 (in the range of redshifts z ~ 0-0.8) was determined. The use of a larger number of spectra allows a factor ~5 improvement on previous constraints based on the same method. On the whole, we find no evidence of changes in α on such cosmological timescales. The mean variation compatible with our results is 1/langtrangΔα/α = (+0.7 ± 0.7) × 10-14 yr-1. The analysis was extended to the [Ne III] and [S II] doublets, although their usefulness is limited due to the fact that all these doublets in QSOs tend to be fainter than [O III], and that some of them are affected by systematics.