An experiment was carried out in an acrylic glass-sided re-circulating closed conduit with a rectangular cross section, which is similar in construction to an erosion function apparatus. An adjustable sand box, made of acrylic glass, was attached to the bottom of the conduit as the sand zone or the test section. The hydraulics of the flow in the erosion function apparatus is complicated due to the limited part of the non-smooth and erodible soil surface attached to the closed conduit. As the bed shear stress changes with the bed roughness, even though the flow velocity does not change, establishing a method to estimate the incipient motion is an important challenge for an erosion function apparatus. The present study was conducted to explore the incipient motion of sands from bed shear stress estimated by four different indirect methods on both the sand bed and the smooth bed installed in the erosion function apparatus. In the experiment, particle image velocimetry (PIV) was used to investigate flow dynamics and incipient motion in terms of dimensionless critical bed shear stress. The experimental results show that the bed shear stress estimated from the log-law profiles in the sand zone and the smooth zones are relatively higher than those of the other indirect methods. The dimensionless critical bed shear stress of threshold condition evaluated by all indirect methods was found in good agreement with those of previous results in both zones. The Manning roughness and Darcy–Weisbach friction coefficients were evaluated based on the critical shear velocity at the incipient motion. Although these coefficients were found slightly greater in the smooth zone than in the sand zone, in both zones, they showed good agreement with previous studies.