  A new era of exploring the early Universe may have begun with the recent
strong evidence for the stochastic gravitational wave (GW) background from the
data reported by NANOGrav, EPTA, PPTA, and CPTA. Inspired by this, we propose a
new potential source of stochastic GWs in the minimal supersymmetric standard
model (MSSM), which could be the theory at a very high energy scale. This
source is the "axion" field in the Higgs multiplets when the Higgs field takes
a large value along the D-flat direction in the early Universe, for example,
during inflation. The axion motion triggers the instability of the standard
model ${\rm U}(1)$ and/or ${\rm SU}(3)$ gauge fields, producing stochastic GWs
during the inflation. This scenario can be seen as a simple UV completion of
the commonly studied models where an axion spectator/inflaton is coupled to a
hidden ${\rm U}(1)$ or ${\rm SU}(N)$ gauge field without matter fields. Thus
the nanohertz GWs may be a sign of supersymmetry. Primordial magnetic field
production is also argued. In addition, we point out the simple possibility
that this axion within the MSSM drives inflation.
