The realization of fractional Chern insulators has largely relied on isolated flat bands in complex bilayer or multilayer moiré systems with ideal quantum geometry. Whether such phases can emerge in genuinely dispersive bands, and by what mechanism, remains an open question. Here, we propose a mechanism for magnetic-field-induced strong correlations in the highly dispersive bands of monolayer graphene on hexagonal boron nitride. In the low-field classical-to-quantum crossover regime of the second moiré miniband, we show that the interplay of the moiré potential and weak magnetic fields (B < 1.5 T) profoundly alters electron dynamics. Saddle-point van Hove singularities produce extended open orbits that strongly suppress carrier propagation, potentially favoring a regime in which Coulomb interactions become comparatively more important, while weak magnetic fields lift valley degeneracy and trigonal warping redistributes Berry curvature. High-resolution magnetotransport and temperature-dependent measurements reveal fractional-slope transport features and signatures of interaction-driven incompressibility. Because the Hall conductivity is not fully quantized and the longitudinal resistance remains finite, we describe these observations as incipient FCI behavior: transport signatures consistent with a developing fractional Chern insulating state, without claiming a fully developed FCI phase established by complete Hall quantization and vanishing longitudinal resistance. Our results suggest a possible fermiology-driven route toward correlated topological states in dispersive bands in the classical-to-quantum crossover, distinct from the conventional flat-band scenario.