The nuclear collective response at finite temperature is investigated for the first time in the quantum framework of the small amplitude limit of the extended time-dependent Hartree-Fock approach, including a non-Markovian collision term. It is shown that the collision width satisfies a secular equation. By employing a Skyrme force, the isoscalar monopole, isovector dipole, and isoscalar quadrupole excitations in 40Ca are calculated and important quantum features are pointed out. The collisional damping due to decay into incoherent two-particle–two-hole states is small at low temperatures but increases rapidly at higher temperatures.