  We present a table-top quantum estimation protocol to measure the
gravitational acceleration $g$ by using an optomechanical cavity. In
particular, we exploit the non-linear quantum light-matter interaction between
an optical field and a massive mirror acting as mechanical oscillator. The
gravitational field influences the system dynamics affecting the phase of the
cavity field during the interaction. Reading out such phase carried by the
radiation leaking from the cavity, we provide an estimate of the gravitational
acceleration through interference measurements. Contrary to previous studies,
having adopted a fully quantum description, we are able to propose a quantum
analysis proving the ultimate bound to the estimability of the gravitational
acceleration and verifying optimality of homodyne detection. Noticeably, thanks
to the light-matter decoupling at the measurement time, no initial cooling of
the mechanical oscillator is in principle demanded.
