  In this paper, we use quasars calibrated from type Ia supernova (SN Ia) to
constrain cosmological models. We consider three different X-ray luminosity
($L_{X}$) - ultraviolet luminosity ($L_{UV}$) relations of quasars, i.e., the
standard $L_{X}$-$L_{UV}$ relation and two redshift-evolutionary relations
(Type I and Type II) respectively constructed from copula and considering a
redshift correction to the luminosity of quasars. Only in the case of the Type
I relation, quasars can always provide effective constraints on the
$\Lambda$CDM model. Furthermore, we show that, when the observational Hubble
data (OHD) are added, the constraints on the absolute magnitude $M$ of SN Ia
and the Hubble constant $H_0$ can be obtained. In the $\Lambda$CDM model, the
OHD measurements plus quasars with the Type I relation yields $M$
=$-19.321^{+0.085}_{-0.076}$, which is in good agreement with the measurement
from SH0ES ($M=-19.253\pm{0.027}$), and $H_0$ =
$70.80\pm3.6~\mathrm{km~s^{-1}Mpc^{-1}}$, falling between the measurements from
SH0ES and the Planck cosmic microwave background radiation data.
