  We present a measurement of the mean absorption of cool gas traced by Mg II
(${\lambda\lambda 2796, 2803}$) around emission line galaxies (ELGs), spanning
spatial scales from 20 kpc to 10 Mpc. The measurement is based on
cross-matching the positions of about 2.5 million ELGs at $z = 0.75-1.65$ and
the metal absorption in the spectra of 1.4 million background quasars with data
provided by the Year 1 sample of the Dark Energy Spectroscopic Instrument
(DESI). The ELGs are divided into two redshift intervals: $0.75 < z < 1.0$ and
$1.0 < z < 1.65$. We find that the composite spectra constructed by stacking
the ELG-QSO pairs show evolution with redshift, with $z>1$ having a
systematically higher signal of Mg II absorption. Within 1 Mpc, the covering
fraction of the cool gas at $z > 1$ is higher than that of $z < 1$. The
enhancement becomes less apparent especially if the projected distance
$r_{p}>$1 Mpc. Also, ELGs with higher stellar mass and star formation rate
(SFR) yield higher clustering of Mg II absorbers at $z<1$. For $z>1$, the
covering fractions with different SFRs show little difference. The higher Mg II
absorption at higher redshift also supports the observations of higher star
formation at cosmic noon. Besides, the profile of Mg II absorption reveals a
change of slope on scales of about 1 Mpc, consistent with the expected
transition from a dark matter halo-dominated environment to a regime where
clustering is dominated by halo-halo correlations. We estimate the cool gas
density profile and derive the metal abundance at different redshifts. The
growth of metal abundance suggests an increased presence of cool gas in the
intergalactic medium (IGM) towards higher redshifts.
