grule = {g[r] -> -((6 f[r])/r^2) + (2 D[f[r], r])/r + D[f[r], r, r]}

A = Laplacian[f[r] (3 Cos[theta]^2 - 1), {r, theta, phi},
        "Spherical"] // ExpandAll // Collect[#, {f[r], f'[r], f''[r]}] &

A = A /. Sin[theta]^2 -> 1 - Cos[theta]^2

(Coefficient[A, f'[r]] // Expand)/(2/r) // Expand
(*3 Cos[theta]^2-1*)
(Coefficient[A, f[r]] // Expand)/(-6/r^2) // Expand
(Coefficient[A, f[r]] // Expand)/(-6/r^2) // Expand

B = g[r] (3 Cos[theta]^2 - 1) /. grule

A - B // FullSimplify
(*0*)