  In this paper, we investigate the evolution of strange quark stars (SQS) from
birth as proto-strange quark stars to maturity as stable SQSs at a zero
temperature. We assume that self-bound free quarks form {entirely the compact}
star and study its evolution {through a series of snapshots} using a
density-dependent quark mass model. We consider $\beta$-equilibrated stellar
matter at two major stages of the star's evolution: neutrino trapped regime and
neutrino transparent regime during the deleptonization and cooling processes of
the star. We fix the entropy density per baryon and the lepton fraction to
investigate the nuclear equation of state (EoS), particle distribution,
temperature profile inside the star, sound velocity, polytropic index, and the
structure of the star. Our results show that stars with higher neutrino
concentrations are slightly more massive than the neutrino-poor ones along the
evolution lines of the SQS. We obtain EoSs in agreement with the conformal
boundary set through sound velocity, and also the 2 M$_\odot$ mass constraint
for NSs was satisfied at all stages of the star's evolution.
