  The $(g-2)_\mu$ anomaly indicates that the second generation of leptons
should have new interactions beyond the standard model. The high flux of
$\nu_\mu$ and $\bar{\nu}_\mu$ at the forward experiments such as FASER$\nu$ and
SND@LHC makes them suitable setups to search for new interactions of the second
generation leptons. In this paper, we build a model in which the second
generation left-handed leptons couple to a new right-handed neutrino, $N$ and a
new Higgs doublet which also couples to the quarks. The scattering of high
energy $\nu_\mu$ off nuclei can produce $N$. We investigate how forward
experiments can test this model by looking for the $N$ production vertex
followed by the displaced vertex of the $N$ decay. Discovering even a single
such event can be a harbinger to look for the spectacular signals of the new
Higgs doublet production at the LHC. We discuss the possibility of explaining
the $(g-2)_\mu$ anomaly by adding more generations of $N$ which will lead to
chain decays of $N$ and multiple leptons with distinct signals both at forward
experiments and at the CMS and ATLAS detectors. Finally, we show that by adding
a new light singlet scalar mixed with the neutral component of the new Higgs
doublet ({\it i.e.,} 2HDM+$S$ model), the statistics of the data sample can be
dramatically increased.
