We present a detailed study of the magnetic properties of weakly ferromagnetic/quantum critical Nb1-yFe2+y using muon spin rotation and relaxation (μSR). By means of an angular dependent study of the muon spin rotation signal in applied magnetic fields on a single crystal in the paramagnetic state we establish the muon stopping site in the crystallographic lattice of NbFe2. With this knowledge we develop models to describe the muon spin rotation and relaxation signals in the weakly ferromagnetic, spin density wave, and quantum critical phases of Nb1-yFe2+y and fit the corresponding experimental data. In particular, we quantify the μSR response for quantum critical behavior in Nb1.0117Fe1.9883 and extract the influence of residual weak structural disorder. From our analysis, Nb1-yFe2+y emerges to be uniquely suited to study quantum criticality close to weak itinerant ferromagnetic order.