We present a detailed study of the magnetic properties of weakly ferromagnetic/quantum critical Nb_1-yFe_2+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 Nb_1-yFe_2+y and fit the corresponding experimental data. In particular, we quantify the μSR response for quantum critical behavior in Nb_1.0117Fe_1.9883 and extract the influence of residual weak structural disorder. From our analysis, Nb_1-yFe_2+y emerges to be uniquely suited to study quantum criticality close to weak itinerant ferromagnetic order.