We report measurements of the branching fractions of singly Cabibbo-suppressed decays Λc+→pη and Λc+→pπ0 using the full Belle data sample corresponding to an integrated luminosity of 980.6 fb-1. The data were collected by the Belle detector at the KEKB e+e- asymmetric energy collider. A clear Λc+ signal is seen in the invariant mass distribution of pη. The fitted number of signal events of the Λc+→pη process is 7734±263; from this, we measure the ratio of branching fractions B(Λc+→pη)/B(Λc+→pK-π+)=[2.258±0.077(stat)±0.122(syst)]×10-2, from which we infer the branching fraction B(Λc+→pη)=[1.42±0.05(stat)±0.11(syst)]×10-3. In addition, no significant signal for Λc+→pπ0 is found, so an upper limit on the branching fraction of B(Λc+→pπ0)<8.0×10-5 at a 90% credibility level is set, more than 3 times better than the best current upper limit.