Y[l_,m_,θ_, ϕ_] := SphericalHarmonicY[l,m,θ,ϕ]

H:=-ℏ^2/(2 Z)(1/Sin[θ] D[(Sin[θ]D[#,θ]),θ] + 1/Sin[θ]^2 D[#,{ϕ,2}])&

Integrate[Conjugate[Y[1,0,θ,ϕ]]  H@Y[1,0,θ,ϕ] Sin[θ], {θ,0,Pi}, {ϕ,0,2 Pi}]