Hybrid circuit quantum electrodynamics (QED) involves the study of coherent quantum physics in solid state systems via their interactions with superconducting microwave circuits. Here we present an implementation of a hybrid superconducting qubit that employs a carbon nanotube as a Josephson junction. We realize the junction by contacting a carbon nanotube with a superconducting Pd/Al bi-layer, and implement voltage tunability of the qubit frequency using a local electrostatic gate. We demonstrate strong dispersive coupling to a coplanar waveguide resonator via observation of a resonator frequency shift dependent on applied gate voltage. We extract qubit parameters from spectroscopy using dispersive readout and find qubit relaxation and coherence times in the range of $10-200~\rm{ns}$.