The ability to detect and distinguish quantum interference signatures is important for both fundamental research and for the realization of devices such as electron resonators(1), interferometers(2) and interference-based spin filters(3). Consistent with the principles of subwavelength optics, the wave nature of electrons can give rise to various types of interference effects(4), such as Fabry-Perot resonances(5), Fano resonances(6) and the Aharonov-Bohm effect(7). Quantum interference conductance oscillations(8) have, indeed, been predicted for multiwall carbon nanotube shuttles and telescopes, and arise from atomic-scale displacements between the inner and outer tubes(9,10). Previous theoretical work on graphene bilayers indicates that these systems may display similar interference features as a function of the relative position of the two sheets(11,12). Experimental verification is, however, still lacking. Graphene nanoconstrictions represent an ideal model system to study quantum transport phenomena(13-15) due to the electronic coherence(16) and the transverse confinement of the carriers(17). Here, we demonstrate the fabrication of bowtie-shaped nano-constrictions with mechanically controlled break junctions made from a single layer of graphene. Their electrical conductance displays pronounced oscillations at room temperature, with amplitudes that modulate over an order of magnitude as a function of subnanometre displacements. Surprisingly, the oscillations exhibit a period larger than the graphene lattice constant. Charge-transport calculations show that the periodicity originates from a combination of the quantum interference and lattice commensuration effects of two graphene layers that slide across each other. Our results provide direct experimental observation of a Fabry-Perot-like interference of electron waves that are partially reflected and/or transmitted at the edges of the graphene bilayer overlap region.