  The exact solutions of the Dirac equation that describe a massive,
non-charged particle with spin-(1/2) in the curved spacetime geometry of
regular Bardeen black hole surrounded by quintessence (BBHSQ) are investigated.
We first derive the Dirac equation in the BBHSQ background using a null tetrad
in the Newman-Penrose formalism. Afterward, we separate the Dirac equation into
ordinary differential equations for the radial and angular parts. The angular
equations are solved exactly in terms of standard spherical harmonics. The
radial part equations are transformed into a Schr\"odinger like differential
wave equations with effective potentials. The effect of the quintessence on the
regular Bardeen black hole is studied by understanding the physical behavior of
the effective potentials. In addition, the potentials are plotted with changing
the quintessence parameters, magnetic monopole charge parameter and the
frequency of the particle in the physically acceptable regions. Finally, we
study the greybody radiation of spin-0 particles from the BBHSQ.
