We report on the Shubnikov--de Haas (SdH) oscillations in the quasi-two-dimensional molecular conductor $\ensuremath{\alpha}\text{\ensuremath{-}}{(\mathrm{BETS})}_{2}{\mathrm{I}}_{3}$ [BETS: bis(ethylenedithio)tetraselenafulvalene] laminated on polyimide films at 1.7 K. From the SdH phase factor we verified experimentally that the material is in the Dirac fermion phase under pressure. $\ensuremath{\alpha}\text{\ensuremath{-}}{(\mathrm{BETS})}_{2}{\mathrm{I}}_{3}$ is in the vicinity of the phase transition between strongly correlated insulating and Dirac fermion phases, and is a possible candidate for an ambient-pressure molecular Dirac fermion system. However, the SdH oscillations indicate that the Berry phase is zero at ambient pressure. Under pressure, a $\ensuremath{\pi}$ Berry phase emerges when the metal-insulator crossover is almost suppressed at $\ensuremath{\sim}0.5$ GPa. The results contrast with those for the pioneering molecular Dirac fermion system $\ensuremath{\alpha}\text{\ensuremath{-}}{(\mathrm{BEDT}\text{\ensuremath{-}}\mathrm{TTF})}_{2}{\mathrm{I}}_{3}$ [BEDT-TTF: bis(ethylenedithio)tetrathiafulvalene] in which Dirac fermions and semiconducting behavior are simultaneously observed.