A β-NMR study was performed on a Sr₂RuO₄ crystal in the metallic state using a beam of spin-polarized ⁸Li⁺ implanted at a mean depth of 90 nm. The ⁸Li⁺ spin-lattice relaxation rate is strongly influenced by the onset of incommensurate spin fluctuations. The nuclear relaxation rate can be explained using previously published bulk ¹⁷O NMR and inelastic neutron spectroscopy measurements of the dynamic magnetic susceptibility to model the hyperfine coupling. A well-resolved quadrupolar-split NMR for ⁸Li⁺ implies a static stopping position in an interstitial site. The ⁸Li⁺ Knight shift is highly sensitive to the anisotropic static susceptibility.