  We study the $f(Q, T)$ gravity in the framework of Weyl geometry (known as
Weyl-type $f(Q, T)$ gravity), where $Q$ denotes the non-metricity scalar, and
$T$ denotes the energy-momentum tensor trace. In this work, we consider the
$f(Q,T)$ model, which is defined as $f(Q,T)=\alpha
Q^{m+1}+\frac{\beta}{6\kappa^2}T$ and investigating two scenarios: $(I)$ $m=0$
(linear model) and $(II)$ $m\neq 0$ (nonlinear model). For both scenarios, we
find the explicit solution for the field equations by using the barotropic
equation of state as $p=\omega\rho$, where $\omega$ is the equation-of-state
(EoS) parameter. Further, we study the obtained solutions statistically using
the $Pantheon^+$ dataset with 1701 data points. For both models, the best-fit
values of model parameters for $1-\sigma$ and $2-\sigma$ confidence level are
mentioned in table 1 and 2. We statistically compare our models to the
$\Lambda$CDM model using ${{\chi}^2_{min}}$, ${{\chi}^2_{red}}$, $AIC$ and
$\Delta AIC$. We also examined the current values of cosmological parameters
such as deceleration and EoS parameters to determine the current acceleration
expansion of the universe. Furthermore, we test our model using $Om$ diagnostic
and compare it to the $\Lambda$CDM model to determine its dark energy profile.
Finally, we draw the conclusion that, statistically speaking, Weyl $f(Q,T)$
model I (linear model) is more compatible with the $\Lambda$CDM model than Weyl
$f(Q,T)$ model II (nonlinear model).
