The intrinsically weak nonlinear optical response of existing materials, further constrained by symmetry-forbidden second-order processes in centrosymmetric media, severely limits efficient frequency conversion in deeply subwavelength, ultrathin volumes. Addressing this challenge is crucial for the development of nonlinear nanophotonics. Here, we show that atomically thin, epitaxially grown crystalline silver films circumvent these restrictions through the interplay of vertical electronic quantum confinement and lateral plasmonic enhancement. We fabricate atomically thin films that exhibit an enhanced nonlinear response associated with electronic quantum wells, and subsequently pattern them into periodic nanoribbon and nanotriangle arrays sustaining infrared localized surface plasmon resonances. Strong near-field confinement in these structures further boosts second-harmonic generation compared to unpatterned films. Precise control over nanostructure geometry enables spectral tuning of the plasmonic resonance, and consequently, the enhanced harmonic frequency. Our findings establish an approach for activating robust second-order nonlinearities in quantum-confined metals, where intrinsic size effects and plasmonic resonances act synergistically. The compatibility of high-quality epitaxial growth with microchip fabrication technology offers a scalable route toward ultracompact nonlinear optical components for on-chip frequency conversion, sensing, and quantum photonic applications.