  Zero-field $\mu$SR measurements in $^{63}$Cu isotope enriched and natural
YBa$_2$Cu$_4$O$_8$ powders are presented. The temperature dependence of the
$\mu^+$ relaxation rate is characterized by a sizeable enhancement below T$_c$.
The comparison of the asymmetry decay in the two samples reveals that the
$\mu^+$ relaxation is driven by nuclear dipole interaction from 300 K down to
4.2 K. It is argued that the increase in the relaxation below T$_c$ originates
from a change in $\mu^+$ site, possibly due to a modification in the charge
distribution within CuO chains.
