We have performed magneto-resistance measurements for single crystals of URu₂Si_2-xP_x (x = 0, 0.006, 0.01, 0.02, and 0.102) in magnetic fields up to 45 T and down to 70 mK. Measurements were performed with the angle between the field and crystalline axes varied from θ = 0° - 120°, where θ = 0° for H // [100] and θ = 90° for H // [001]. Quantum oscillations are detected up to x = 0.02 and are most visible when H is nearly parallel to [100] owing to the large anisotropy of the field induced ordered phases. From this, we are able to describe the evolution of the Fermi surface topology with electron doping as hidden order collapses towards the paramagnetic state at low temperatures. At two angles, θ = 76° and 90°, temperature sweeps were performed up to ~700 mK. From Lifshiftz-Kosevich fits to the quantum oscillations amplitudes we find that the substitution dependent quasiparticle effective mass μ_eff displays a maximum at x = 0.01, consistent with earlier evidence from heat capacity measurements for an unexpected electronic instability in this x-range.