  In response to recent proton spin relaxation-time measurements on
NiCu(pba)(H$_2$O)$_3$$\cdot$2H$_2$O with ${pba}=1,3{-propylenebis(oxamato)}$,
which is an excellent one-dimensional ferrimagnetic Heisenberg model system of
spin-$(1,{1/2})$, we study the Raman relaxation process in spin-$(S,s)$ quantum
ferrimagnets on the assumption of predominantly dipolar hyperfine interactions
between protons and magnetic ions. The relaxation time $T_1$ is formulated
within the spin-wave theory and is estimated as a function of temperature and
an applied field $H$ by a quantum Monte Carlo method. The low-temperature
behavior of the relaxation rate $T_1^{-1}$ qualitatively varies with $(S,s)$,
while $T_1^{-1}$ is almost proportional to $H^{-1/2}$ due to the characteristic
dispersion relations.
