We report quantum interference effects in the electrical conductance of chemical vapour deposited graphene nanoconstrictions fabricated using feedback controlled electroburning. The observed multi-mode Fabry-Pérot interferences can be attributed to reflections on potential steps inside the channel. Sharp anti-resonance features with a Fano line shape are observed. Theoretical modelling reveals that these Fano resonances are due to localised states inside the constriction, which couple to the delocalised states that also give rise to the Fabry-Pérot interference patterns. This study provides new insight into the interplay between two fundamental forms of quantum interference in graphene nanoconstrictions.