  In this article, we attempt to explore the dark sector of the universe i.e.
dark matter and dark energy, where the dark energy components are related to
the modified $f(Q)$ Lagrangian, particularly a power law function $f(Q)= \gamma
\left(\frac{Q}{Q_0}\right)^n$, while the dark matter component is described by
the Extended Bose-Einstein Condensate (EBEC) equation of state for dark matter,
specifically, $p = \alpha \rho + \beta \rho^2$. We find the corresponding
Friedmann-like equations and the continuity equation for both dark components
along with an interacting term, specifically $\mathcal{Q} = 3b^2H \rho$, which
signifies the energy exchange between the dark sector of the universe. Further,
we derive the analytical expression of the Hubble function, and then we find
the best-fit values of free parameters utilizing the Bayesian analysis to
estimate the posterior probability and the Markov Chain Monte Carlo (MCMC)
sampling technique corresponding to CC+Pantheon+SH0ES samples. In addition, to
examine the robustness of our MCMC analysis, we perform a statistical
assessment using the Akaike Information Criterion (AIC) and Bayesian
Information Criterion (BIC). Further from the evolutionary profile of the
deceleration parameter and the energy density, we obtain a transition from the
decelerated epoch to the accelerated expansion phase, with the present
deceleration parameter value as $q(z=0)=q_0=-0.56^{+0.04}_{-0.03}$ ($68 \%$
confidence limit), that is quite consistent with cosmological observations. In
addition, we find the expected positive behavior of the effective energy
density. Finally, by examining the sound speed parameter, we find that the
assumed theoretical $f(Q)$ model is thermodynamically stable.
