The Majorana zero modes in vortex cores are of extensive interest in the context of topological quantum computing. However, a zero-energy bound state may arise accidentally but is not necessarily a Majorana zero mode. Such accidental zero modes should be carefully ruled out in experiment in order to identify the genuine Majorana zero modes. We show that in a spin-orbital coupled multi-band superconductor, such as the iron-selenide superconductor, accidental zero modes indeed arise in the vortex core if the pairing symmetry is the so-called nodeless d-wave (defined in the absence of spin-orbital coupling). Instead, if the pairing symmetry is s₊₊ or s± with respect to the Fermi pockets split by the spin-orbital coupling, the accidental zero modes do not appear in the limit of weak spin-orbital coupling. Our results are not only important in the experimental identification of Majorana zero modes, but also provide an avenue to pinpoint the pairing symmetry of the iron-selenide superconductor.