  In this paper, we investigate some exact cosmological models in Myrzakulov
$F(T,Q)$ gravity or the Myrzakulov gravity-III (MG-III) proposed in
[arXiv:1205.5266], with observational constraints. The MG-III gravity is some
kind of unification of two known gravity theories, namely, the $F(T)$ gravity
and the $F(Q)$ gravity. The field equations of the MG-III theory are obtained
by regarding the metric tensor and the general affine connection as independent
variables. We then focus on the particular case in which the $F(T,Q)$ function
characterizing the aforementioned metric-affine models is linear that is
$F(T,Q)=\lambda T+\mu Q$. We investigate this linear case and consider a
Friedmann-Lema\^{i}tre-Robertson-Walker background to study cosmological
aspects and applications. We have obtained three exact solutions of the
modified field equations in different cases $T$ and $Q$, in the form of Hubble
function $H(t)$ and scale factor $a(t)$ and placed observational constraints on
it through the Hubble $H(z)$ datasets on it using the MCMC analysis. We have
investigated the deceleration parameter $q(z)$, effective EoS parameters and a
comparative study of all three models with $\Lambda$CDM model has been carried
out.
