  The Inert Higgs Doublet model (IDM), assisted by Peccei-Quinn (PQ) symmetry,
offers a simple but natural framework of a dark sector that accommodates Weakly
Interacting Massive Particle (WIMP) and axion as dark matter components.
Spontaneous breaking of $U(1)_{PQ}$ symmetry, which was originally proposed as
an elegant solution to the strong charge-parity (CP) problem, also ensures the
stability of WIMP through a residual $\mathbb{Z}_2$ symmetry. Interestingly,
additional fields necessitated by PQ symmetry further enrich the dark sector.
These include a scalar field proprietor for axion DM and a vector-like quark
(VLQ) that acts as a portal for the dark sector through Yukawa interactions.
Moreover, this combination of the axion and WIMP components satisfies the
observed DM relic density and reopens the phenomenologically exciting region of
the IDM parameter space where the WIMP mass falls between 100 - 550 GeV. We
investigate the model-independent pair production of VLQs exploring this region
at the Large Hadron Collider (LHC), incorporating the effects of
next-to-leading order (NLO) QCD corrections. After production, each VLQ decays
into a top or bottom quark accompanied by an inert scalar, a consequence of the
residual $\mathbb{Z}_2$ symmetry. Utilising relevant observables with a
leptonic search channel and employing multivariate analysis, we demonstrate the
ability of this analysis to exclude a significant portion of the parameter
space with an integrated luminosity of 300 $\text{fb}^{-1}$.
