In this work we investigate the dynamic properties of hemoglobin in glycerolD(8)/D(2)O solution using incoherent elastic (ENS) and quasi-elastic (QENS) neutron scattering. Taking advantage of complementary energy resolutions of backscattering spectrometers at ILL (Grenoble), we explore motions in a large space-time window, up to 1 ns and 14 A; moreover, in order to cover the harmonic and anharmonic protein dynamics regimes, the elastic experiments have been performed over the wide temperature interval of 20-300 K. To study the dependence of the measured dynamics upon the protein quaternary structure, both deoxyhemoglobin (in T quaternary conformation) and carbonmonoxyhemoglobin (in R quaternary conformation) have been investigated. From the ENS data the mean square displacements of the non-exchangeable hydrogen atoms of the protein and their temperature dependence are obtained. In agreement with previous results on hydrated powders, a dynamical transition at about 220 K is detected. The results show interesting differences between the two hemoglobin quaternary conformations, the T-state protein appearing more rigid and performing faster motions than the R-state one; however, these differences involve motions occurring in the nanosecond time scale and are not detected when only faster atomic motions in the time scale up to 100 ps are investigated. The QENS results put in evidence a relevant Lorentzian quasi-elastic contribution. Analysis of the dependence of the Elastic Incoherent Structure Factor (EISF) and of the Lorentzian halfwidth upon the momentum transfer suggests that the above quasi-elastic contribution arises from the diffusion inside a confined space, values of confinement radius and local diffusion coefficient being compatible with motions of hydrogen atoms of the amino acid side chains. When averaged over the whole range of momentum transfer the QENS data put in evidence differences between deoxy and carbonmonoxy hemoglobin and confirm the quaternary structure dependence of the protein dynamics in the nanosecond time scale.