We investigate electron transport inside a ring system composed of a quantum dot (QD) coupled to two Majorana bound states confined at the ends of a one-dimensional topological superconductor nanowire. By tuning the magnetic flux threading through the ring, the model system we consider can be switched into states with or without zero-energy modes when the nanowire is in its topological phase. We find that the Fano profile in the conductance spectrum due to the interference between bound and continuum states exhibits markedly different features for these two different situations, which consequently can be used to detect the Majorana zero-energy mode. Most interestingly, as a periodic function of magnetic flux, the conductance shows 2π periodicity when the two Majorana bound states are nonoverlapping (as in an infinitely long nanowire) but displays 4π periodicity when the overlapping becomes nonzero (as in a finite length nanowire). We map the model system into a QD–Kitaev ring in the Majorana fermion representation and affirm these different characteristics by checking the energy spectrum.