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Monitoring mass accretion onto substellar objects provides insights into the geometry of the accretion flows. We use the Lulin One-meter Telescope to monitor H α emission from FU Tau B, a ∼19 M _Jup brown-dwarf companion at 5.″7 (719 au) from the host star, for six consecutive nights. This is the longest continuous H α monitoring for a substellar companion near the deuterium-burning limit. We aim to investigate if accretion near the planetary regime could be rotationally modulated as suggested by magnetospheric accretion models. We find tentative evidence that H α mildly varies on hourly and daily timescales, though our sensitivity is not sufficient to definitively establish any rotational modulation. No burst-like events are detected, implying that accretion onto FU Tau B is overall stable during the time baseline and sampling windows over which it was observed. The primary star FU Tau A also exhibits H α variations over timescales from minutes to days. This program highlights the potential of monitoring accretion onto substellar objects with small telescopes.