We study an effective quantum cosmological model with a scale-dependent gravitational coupling $G\left(a\right)$. Starting from a Newtonian cosmological construction inspired by a corrected two-body gravitational potential, we derive an effective Friedmann-type equation and quantize the corresponding minisuperspace model to obtain a Wheeler–DeWitt equation. For the power-law running $G\left(a\right)=G_0{a}^n$, the small-scale behavior is controlled by the exponent n. In the flat dust model under the repulsive effective-potential prescription adopted here, with a quantum correction term $\mathrm{\Xi }>0$, the values $n=3/2$ and $n=2$ mark distinct analytic boundaries: $n=3/2$ marks the sign change of the quantum-correction coefficient, while $n=2$ marks the degeneracy of the leading powers of a. A genuine bounce occurs only when a root of $H^2\left(a\right)$ separates a forbidden small-a region from an allowed expanding branch, so that nonsingular sectors appear without spatial curvature, exotic matter, or nonstandard quantization prescriptions. We obtain Bessel, Airy, numerical, and WKB solutions, and show that the model separates genuine DeWitt suppression from regularity at the origin. The WKB analysis exhibits a forbidden region, a turning point, and an oscillatory semiclassical branch, while the conserved probability current distinguishes expanding and contracting sectors. A scalar field treated as a weakly backreacting internal clock yields wave packets that track the leading-order classical relational trajectory at a large scale factor; the WKB Hamilton–Jacobi limit recovers the modified Friedmann dynamics as the leading-order phase of the Wheeler–DeWitt wave function.