We study the emergence of Majorana zero modes (MZMs) at the ends of a finite double zigzag honeycomb nanoribbon (zHNR), considering a minimal model composed of the first nearest neighbor hopping term, Rashba spin-orbit coupling (RSOC), p-wave superconducting pairing, and an applied external magnetic field (EMF). The energy spectrum profiles reveal regions with either spin up or down MZMs belonging to distinct topological phase transitions, which are characterized by their corresponding winding numbers and can be accessed by tuning the chemical potential of the nanoribbons. Hybrid systems constituted by zHNRs deposited on conventional s-wave superconductors are potential candidates for experimentally realizing our proposal. The spin discrimination of MZMs suggests a possible route for performing topological-conventional qubit operations using Majorana spintronics.