We study tetravalent analogues to fullerene systems which include Goldberg polyhedra (genus 0) and toroidal polyhedra (genus 1), where each node is connected to four others. According to the Euler-Poincaré formula, a tetravalent polyhedron with genus 0 has exactly 8 triangles, while on a toroidal polyhedron, the number of triangles and pentagons must be equal. We develop a construction method for toroidal polyhedra using a methodology similar to our previous work, which categorizes tetravalent toroidal polyhedra with a set of five indices. Bead models of the tori are presented as well.