The real-time quantum dynamics of a series of lithium para-hydrogen clusters, Li(p-H-2)(n) (n = 13, 55, and 180), has been investigated at 2.5 and 4.0 K by means of normal mode centroid path integral molecular dynamics (NMCMD) simulation, following the methodology originally proposed by Cao and Voth [J. Chem. Phys. 101, 6168 (1994)]. The Li(p-H-2)(34) and neat (p-H-2)(34) clusters have also been simulated at 2.5 K to see the effect of doping of a Li atom on the cluster dynamics. We explicitly display both the microcanonical and the constant-temperature equations of motion for NMCMD simulations using the Nose-Hoover chain thermostats and the reference system propagator algorithm (RESPA). In addition to the energetic and structural properties, the real-time semi-classical dynamics of the centroids of the Li atom and p-H-2 molecules in the clusters has been explored to investigate the diffusive and vibrational properties. In general, quantization of the nuclear motion enhances the ease of melting and diffusion, and also causes the power spectra of the velocity autocorrelation functions of hydrogen molecules to be dramatically shifted to lower frequency than in the classical regime. The self-diffusion coefficient of p-H-2 molecules in the Li(p-H-2)(13) cluster, which is melted completely at both 2.5 and 4.0 K, is comparable to the experimental value of bulk liquid p-H-2 just above the melting temperature (14 K). No significant differences have been found for the structural properties, molecular diffusion, and the quantum vibrational spectra of p-H-2 molecules between the neat (p-H-2)(34) and Li(p-H-2)(34) clusters at 2.5 K. The n = 34, 55, and 180 clusters show a solidlike core and a tendency towards surface melting at 2.5 K, with diffusion restricted to the outer region. The core of the Li(p-H-2)(55) cluster exhibits slow liquidlike diffusion at 4.0 K, whereas that of the Li(p-H-2)(180) cluster remains solidlike at this temperature. The quantum power spectrum of the velocity autocorrelation function of the core molecules in the Li(p-H-2)(180) cluster reveals vibrational frequencies in good accord with experimental neutron and Raman data on the bulk solid. The power spectrum of the Li atom on the cluster surface is peaked at low frequency (<20 cm(-1)).