  In this paper, we analyze parameter regions that can alleviate the Hubble
tension in the U(1)$_{L_\mu - L_\tau}$ model with the broken lepton number
U(1)$_L$ symmetry. As new particles, this model has a U(1)$_{L_\mu - L_\tau}$
gauge boson $Z'$ and a Majoron $\phi$, which can affect the early universe and
the effective number of neutrino species $N_{\rm eff}$. If $Z'$ and $\phi$
simultaneously exist in the early universe, $Z'\,$-$\,\phi$ interaction
processes such as $Z'\nu_\alpha \leftrightarrow \phi\bar{\nu}_\beta$ occur. The
comparison of $N_{\rm eff}$ between the cases with and without the
$Z'\,$-$\,\phi$ interaction processes shows that these processes make a small
contribution of $\mathcal{O}(10^{-4})$ to $N_{\rm eff}$, and it does not need
to be considered for the alleviation of the Hubble tension. Based on these
facts, we calculated $N_{\rm eff}$ for various Majoron parameters without the
$Z'\,$-$\,\phi$ interaction processes to search parameters that could alleviate
the Hubble tension. As a result, we found that the U(1)$_{L_\mu - L_\tau}$
gauge boson and Majoron can alleviate the Hubble tension in some parameter
regions, and there is a non-trivial synergy contribution between $Z'$ and
$\phi$. Moreover, the parameter region with a lighter mass $m_\phi \lesssim 2$
MeV and a larger coupling $\lambda \gtrsim 10^{-8}$ is excluded because it
predicts too large $N_{\rm eff}$, i.e. $N_{\rm eff} \gtrsim 3.5$. The favored
and restricted regions of the Majoron parameters depend on the $Z'$ parameters
because of the presence of the $Z'$ contribution and synergy one.
