We establish a rigorous mathematical framework connecting graphene nanoribbon quantum sensing to the Lambert W function through the finite square well (FSW) analogy. The Lambert W function, defined as the inverse of f(W)=We^W, provides exact analytical solutions to transcendental equations governing quantum confinement. Operating near the branch point singularity at z=-1/e yields sensitivity enhancement factors scaling as (z-z_c)^(-1/2), achieving 35-fold enhancement when the operating point lies within δ=0.001 of the branch point. Comprehensive numerical verification confirms: (i) all seven bound states for strength parameter R=10 satisfy the constraint u²+v²=R² to machine precision; (ii) the theoretical band gap formula E_g=2πℏ v_F/(3W) is analytically equivalent to the independently determined empirical relation E_g=1.38/W~eV·nm, establishing the validity of the FSW-GNR analogy; (iii) a universal sensitivity factorization S_X = Gₖ · η_(rm enh) · P_X applies across biomedical (SARS-CoV-2, inflammatory markers, cancer biomarkers), environmental (CO₂, CH₄, NO₂, N₂O, H₂O), and physical (strain, magnetic field, temperature) sensing modalities. This unified framework provides analytically predictable design principles for next-generation graphene quantum sensors. The framework is analytic and predictive rather than microscopic or experimental: band-structure and adsorption parameters are taken as inputs from tight-binding, first-principles, and experimental sources, and the framework returns closed-form sensitivity and design relations built upon them. Reported detection limits are labelled throughout as either literature-demonstrated device values or values predicted by the present framework.