We generalise the cosmographic reconstruction programme of Chakraborty, Dunsby & Scherrer to scalar-tensor gravity. Working in the Jordan frame with a general non-minimal coupling $F(\Phi )$, we derive exact closed-form expressions for the potential slope ${\lambda }_U$, the coupling slope ${\lambda }_F$, and their curvature parameters ${\Gamma }_U$ and ${\Gamma }_F$ in terms of the cosmographic parameters $(q,j,s)$, the Planck-mass running ${\alpha }_M=dlnF/dlna$ and its derivatives. We express ${\alpha }_M$ and its derivatives in terms of a set of coefficients $g_n$ describing the time variation of the gravitational constant $G\propto F^{-1}$, casting all four quantities in terms of the time derivatives of $G$, which are in principle observable. We then evaluate the reconstruction against current data, including DESI DR2 baryon acoustic oscillation measurements and Solar-System tests of gravity, propagating the cosmographic uncertainties through to the reconstructed potential. All expressions reduce analytically to those of Chakraborty, Dunsby & Scherrer in the minimally-coupled limit, which we verify symbolically. We find that the potential slope ${\lambda }_U$ is recovered at the few-tenths level while the curvature ${\Gamma }_U$ is essentially unconstrained. We further show that for minimally-coupled quintessence the positivity of the scalar kinetic term is equivalent to $w_{\rm eff,0}\geq-1$, so that the region of cosmographic parameter space closed to it is exactly the phantom region, and that the non-minimal coupling reopens this region through a single coefficient $g_2$ of the gravitational tower. Because Solar-System tests restrict $|g_1|\lesssim {10}^{-3}$ at our benchmarks while the reconstruction is sensitive to $g_2\sim 𝒪(0.1)$, the framework presupposes that ${\alpha }_M$ is passing through zero at the present epoch, as expected on the Damour–Nordtvedt least-coupling attractor.