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In the past few years, the lithium niobate on insulator (LNOI) platform has revolutionized lithium niobate materials, and a series of quantum photonic chips based on LNOI have shown unprecedented performances. Quantum frequency conversion (QFC) photonic chips, which enable quantum state preservation during frequency tuning, are crucial in quantum technology. In this work, we demonstrate a low-noise QFC process on an LNOI nanophotonic platform designed to connect telecom and near-visible bands with sum-frequency generation by long-wavelength pumping. An internal conversion efficiency of 73% and an on-chip noise count rate of 900 counts per second (cps) are achieved. Moreover, the on-chip preservation of quantum statistical properties is verified, showing that the QFC chip is promising for extensive applications of LNOI integrated circuits in quantum information. Based on the QFC chip, we construct an upconversion single-photon detector with the sum-frequency output spectrally filtered and detected by a silicon single-photon avalanche photodiode, demonstrating the feasibility of an upconversion single-photon detector on-chip with a detection efficiency of 8.7% and a noise count rate of 300 cps. The realization of a low-noise QFC device paves the way for practical chip-scale QFC-based quantum systems in heterogeneous configurations.