We study the ground state of the quantum KdV model using the ODE/IM correspondence, which relates spectral data of differential equations to integrable structures in quantum field theory. Focusing on a family of anharmonic oscillators, we analyze their spectral determinants and show how they encode the TQ and QQ relations of quantum KdV. Through asymptotic and WKB analysis, we reveal connections to classical special functions, distribution of their zeros and Bohr-Sommerfeld quantization conditions, providing new insights into the integrable structure and semiclassical behavior of the model.