We propose a model describing the evolution of free electron current density in graphene giving rise to bidimensional wormhole solutions. Based on analogue concepts of General Relativity, we perform the analysis using the difference between curvatures of parallel and antiparallel spins. In such a framework, effective “gravitons” emerges in the form of gauge fields exchanged between electrons. In a plain grapheme system, the curvatures produced by both kinds of spins neutralize each other giving rise to no conduction. However, in the presence of geometrical defects of the graphene sheets, the inequality between curvatures leads to the emergence of current densities and conductivity in a wormhole solution. Depending on the type of defects, the resulting current density can be negative or positive. Possible applications are discussed.