We examine neutral-current quasi-elastic neutrino-nucleus reactions on C-12 and Pb-208 targets. We use the relativistic mean field theory approach to describe the nuclear dynamics. We compute the cross sections for the scattering of 150, 500 and 1000 MeV neutrinos on a C-12 target and study the effect of the strange-quark content of the nucleon which appears in these reactions via the isoscalar weak current. We compare our results with the data of the MiniBooNE experiment for mineral oil (CH2). We also calculate the cross section for the quasi-elastic neutron knockout reaction of 20-60 MeV neutrinos on a Pb-208 target which is relevant to plans to use lead as a target material in future supernova neutrino detectors.