Graphene plasmon resonators with the ability to support plasmonic resonances in the infrared region make them a promising platform for plasmon-enhanced spectroscopy techniques. Here we propose a resonant graphene plasmonic system for infrared spectroscopy sensing that consists of continuous graphene and graphene ribbons separated by a nanometric gap. Such a bilayer graphene resonator can support acoustic graphene plasmons (AGPs) that provide ultraconfined electromagnetic fields and strong field enhancement inside the nano-gap. This allows us to selectively enhance the infrared absorption of protein molecules and precisely resolve the molecular structural information by sweeping graphene Fermi energy. Compared to the conventional graphene plasmonic sensors, the proposed bilayer AGP sensor provides better sensitivity and improvement of molecular vibrational fingerprints of nanoscale analyte samples. Our work provides a novel avenue for enhanced infrared spectroscopy sensing with ultrasmall volumes of molecules.