Integrating orthogonal time frequency space (OTFS) modulation into low-Earth-orbit (LEO) satellite communication systems can significantly enhance the robustness against severe Doppler effects caused by fast time-varying channels. However, since the high-dynamic characteristic of terrestrial-to-satellite links considerably constrains the OTFS frame duration, the inevitable fractional Doppler shifts will result in the emergence of inter-Doppler interference (IDI), thereby degrading channel estimation performance. To tackle this challenge, we develop an efficient correlation information-aided channel estimation scheme based on Zadoff-Chu (ZC) sequences in the presence of fractional Doppler shifts. The proposed scheme enables a fast coarse estimation of channel parameters by leveraging the magnitude distribution regularity of the periodical correlation results of ZC pilot under IDI, and remodels channel estimation as a non-convex constrained least squares optimization problem to obtain the fine channel parameters for further resistance of the inter-path interference. Complexity analysis and simulation results validate that our scheme can attain a high estimation accuracy while achieving substantial reductions in both the peak-to-average power ratio and computational complexity, in comparison to the state-of-the-art ones.