Two-dimensional materials with unique and excellent physical and chemical properties have attracted much attention in recent years. Among the two-dimensional materials, graphene or grapheme-like materials with honeycomb structure can be mainly prepared by the chemical vapor deposition (CVD) method. The key of this method is to select the substrates and control the nucleation and growth process of honeycomb structures. Graphene prepared by CVD contains many structure defects and grain boundaries, which mainly arise from nucleation process. However, the nucleation mechanism of graphene prepared by CVD method is not very clear. In addition, more than ten kinds of metal substrates can be used as substrate materials in CVD methods, such as Cu and Ni, which have nearly always face-centered cubic (FCC) structures and similar functions in the preparation process. In order to better describe the nucleation of graphene and understand the influences of metal substrates, we introduce the structural order parameter into the three-mode phase-field crystal model to distinguish the low-density gas phase from condensed phases. Nucleation processes of graphene on substrates with different symmetries are studied at an atomic scale by using the three-mode phase-field crystal model, which can simulate transitions between highly correlated condensed phases and low-density vapor phases. Simulation results indicate that no matter whether there is a substrate in the nucleation process, firstly gaseous atoms gather to form amorphous transitional clusters, and then amorphous transitional clusters gradually transform into ordered graphene crystals, with continuous accumulation of new gaseous atoms and position adjustment of atoms. In the nucleation process, five membered ring structures act as a transitional function. When grown on the substrate with a good geometric match with the honeycomb lattice, such as (111) plane of FCC metals, the graphene island has small structural defects. However, when grown without a substrate or on the substrate with a bad geometric match, such as (100) plane of FCC metals, the graphene island contains many structural defects and grain boundaries, which are not conducive to the preparation of high quality graphene. Compared with the (100) crystal plane of the tetragonal cell, the (110) crystal plane of the rectangular cell is favorable for the preparation of graphene single crystals with less defects. Therefore, the appropriate metal substrate can promote the nucleation process of graphene and reduce the formation of distortions and defects during the nucleation and growth of graphene.