  Absolute distances from strong lensing can anchor Type Ia Supernovae (SNe Ia)
at cosmological distances giving a model-independent inference of the Hubble
constant ($H_0$). Future observations could provide strong lensing time-delay
distances with source redshifts up to $z\,\simeq\,4$, which are much higher
than the maximum redshift of SNe Ia observed so far. In order to make full use
of time-delay distances measured at higher redshifts, we use quasars as a
complementary cosmic probe to measure cosmological distances at redshifts
beyond those of SNe Ia and provide a model-independent method to determine
$H_0$. In this work, we demonstrate a model-independent, joint constraint of
SNe Ia, quasars, and time-delay distances from strong lensed quasars. We first
generate mock data sets of SNe Ia, quasar, and time-delay distances based on a
fiducial cosmological model. Then, we calibrate the quasar parameters model
independently using Gaussian process (GP) regression with mock SNe Ia data.
Finally, we determine the value of $H_0$ model-independently using GP
regression from mock quasars and time-delay distances from strong lensing
systems. As a comparison, we also show the $H_0$ results obtained from mock SNe
Ia in combination with time-delay lensing systems whose redshifts overlap with
SNe Ia. Our results show that quasars at higher redshifts show great potential
to extend the redshift coverage of SNe Ia and thus enable the full use of
strong lens time-delay distance measurements from ongoing cosmic surveys and
improve the accuracy of the estimation of $H_0$ from $2.1\%$ to $1.3\%$ when
the uncertainties of the time-delay distances are $5\%$ of the distance values.
