Characteristic oscillations of neutron stars (NSs) known as quasinormal modes (QNMs) are dependent on the state of matter inside the star. We construct a set of NS matter equations of state (EOSs) with hadron-quark phase transitions (PTs) combining two models - a relativistic mean-field (RMF) model for nucleonic regime and a mean-field theory of quantum chromodynamics (MFTQCD) for quark regime. We investigate the influence of a possible hadron-quark PT on $f-$mode oscillations, a class of QNM oscillations that contributes the most during gravitational wave (GW) emission from a perturbed NS. A fully general relativistic formalism for studying NS QNMs allows us to probe the influence of phase transition on the GW damping times of the oscillations. We employ constraints from nuclear physics and recent astrophysical observations to narrow down the allowed EOSs through Bayesian inference. We find that resulting EOSs manifest a smooth hadron-quark PT based on the set of constraints imposed and the model employed during Bayesian analysis. Additionally, the mass-radius, $f-$mode frequencies, and corresponding GW damping times vary in the presence of deconfined quark matter to such an extent that future observations are needed for a thorough exploration.