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We probe the dimension-six operators contributing to Higgs production in association with a $Z$ boson at the future high-luminosity electron-positron colliders. Potential constraints on dimension-six operators in the Higgs sector are determined by performing a shape analysis on the differential angular distribution of the Higgs and $Z$ boson decay products. The analysis is performed at the center-of-mass energies of 350 and 500 GeV including a realistic detector simulation and the main sources of background processes. The 68\% and 95\% confidence level upper limits are obtained on the contributing anomalous couplings considering only the decay of the Higgs boson into a pair of $b$-quarks and leptonic $Z$ boson decay. Our results show that angular observables provide a great sensitivity to the anomalous couplings, in particular, at the high-luminosity regime.