  In this article, we investigate scalar field cosmology in the coincident
$f(Q)$ gravity formalism. We calculate the motion equations of $f(Q)$ gravity
under the flat Friedmann-Lema\^{i}tre-Robertson-Walker background in the
presence of a scalar field. We consider a non-linear $f(Q)$ model, particularly
$f(Q)=-Q+\alpha Q^n$, which is nothing but a polynomial correction to the STEGR
case. Further, we assumed two well-known specific forms of the potential
function, specifically the exponential from $V(\phi)= V_0 e^{-\beta \phi}$ and
the power-law form $V(\phi)= V_0\phi^{-k}$. We employ some phase-space
variables and transform the cosmological field equations into an autonomous
system. We calculate the critical points of the corresponding autonomous
systems and examine their stability behaviors. We discuss the physical
significance corresponding to the exponential case for parameter values $n=2$
and $n=-1$ with $\beta=1$, and $n=-1$ with $\beta=\sqrt{3}$. Moreover, we
discuss the same corresponding to the power-law case for the parameter value
$n=-2$ and $k=0.16$. We also analyze the behavior of corresponding cosmological
parameters such as scalar field and dark energy density, deceleration, and the
effective equation of state parameter. Corresponding to the exponential case,
we find that the results obtained for the parameter constraints in Case III is
better among all three cases, and that represents the evolution of the universe
from a decelerated stiff era to an accelerated de-Sitter era via
matter-dominated epoch. Further, in the power-law case, we find that all
trajectories exhibit identical behavior, representing the evolution of the
universe from a decelerated stiff era to an accelerated de-Sitter era. Lastly,
we conclude that the exponential case shows better evolution as compared to the
power-law case.
