[O III]$\lambda\lambda$4960,5008 doublet are often the strongest narrow emission lines in starburst galaxies and quasi-stellar objects (QSOs), and thus are a promising probe to possible variation of the fine-structure constant $\alpha$ over cosmic time. Previous such studies using QSOs optical spectra were limited to $z<1$. In this work, we constructed a sample of 40 spectra of Ly$\alpha$ emitting galaxies (LAEs) and a sample of 46 spectra of QSOs at $1.09Comment: 24 pages, 22 figures. Accepted for publication in MNRAS