Quintessence Universe and cosmic acceleration in $f(Q,T)$ gravity

The problem of cosmic acceleration and dark energy is one of the mysteries presently posed in the scientific society that general relativity has not been able to solve. In this work, we have considered alternative models to explain this late-time acceleration in a flat Friedmann-Lema\^i% tre-Robertson-Walker (FLRW) Universe within the framework of the $$\ modified gravity theory (where $Q$ is the non-metricity and $T$ is the trace of the energy-momentum tensor) recently proposed by Xu et al. (Eur. Phys. J. C 79 (2019) 708), which is an extension of $f\left( Q\right)$ gravity with the addition of the $T$ term. Here, we presume a specific form of $f\left( Q,T\right) =\alpha Q+\beta Q^{2}+\gamma T$ where $$, $\beta$ and $\gamma$ are free model parameters, and obtained the exact solutions by assuming the cosmic time-redshift relation as t(z)=\frac{% nt_{0}}{m}g(z) which produces the Hubble parameter of the form H(z)=\frac{% mH_{0}}{m+n}\left[ \frac{1}{g(z)}+1\right] where $m$ and $n$ are the non-negative constants, we find the best values for them using 57 data points of the Hubble parameter $H\left( z\right)$. Also, we find the behavior of different cosmological parameters as the deceleration parameter $$, energy density $\left( \rho \right)$, pressure $$ and EoS parameter $\left( \omega \right)$ and compare them with the observational results. To ensure the validity of the results, we studied the energy conditions along with jerk parameter. Finally, we find that our model behaves similarly to the quintessence Universe.