  [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
$1.09<z<3.73$ using the VLT/X-Shooter near-infrared spectra publicly available.
We measured the wavelength ratios of the two components of the spin-orbit
doublet and accordingly calculated $\alpha(z)$ using two methods. Analysis on
all of the 86 spectra yielded $\Delta\alpha/\alpha=(-3\pm6)\times10^{-5}$ with
respect to the laboratory $\alpha$ measurements, consistent with no variation
over the explored time interval. If assuming a uniform variation rate, we
obtained $\alpha^{-1}{\rm d}\alpha/{\rm d}t = (-3\pm6)\times10^{-15}$ yr$^{-1}$
within the last 12 Gyrs. Extensive tests indicate that $\alpha$ variation could
be better constrained using starburst galaxies' spectra than using QSO spectra
in future studies.
