Quantum frequency conversion (QFC) enables optical interfaces between dissimilar quantum systems, but suppressing noise while maintaining high conversion efficiency remains challenging. Here, we demonstrate that adapted poling in thin-film lithium niobate (TFLN) waveguides reduces the pump-power requirement while shifting the dominant noise mechanism and its power scaling. In 2-cm-long waveguides, adapted poling enables maximum conversion efficiency at an on-chip pump power of 19±0.8~mW, reducing the required pump power by 8 dB and the total conversion noise by 14 dB relative to conventional periodic poling at their respective maximum-conversion operating points. Controlled measurements with different pump conditions and device configurations reveal a transition from predominantly quadratic noise scaling in periodically poled devices to predominantly linear scaling in adapted-poling devices. This transition accompanies a shift from noise dominated by cascaded second-harmonic generation and Stokes spontaneous Raman scattering to direct anti-Stokes spontaneous Raman scattering. These results establish adapted poling as an effective approach to suppressing nonlinear noise, enabling low-noise integrated frequency converters for quantum networks.