In this article, we consider the first-moment model approximation of the radiative transfer equation. This system is linear hyperbolic and satisfies a diffusion limit. Some finite volume numerical schemes have been proposed which reproduce this diffusion limit [8, 9]. Here, we extend such schemes, originally defined on polygonal meshes, to conical meshes (using rational quadratic Bezier curves). We obtain really new schemes that do not reduce to the polygonal version when the conical edges tend to straight lines. Moreover, these schemes can handle curved unstructured meshes so that geometric error on initial data representation is reduced and geometry of the domain is improved. Extra flux coming from conical edge (through his mid edge point) has a deep impact on the stabilization when compared to the original polygonal scheme. Cross stencil phenomenon of polygonal scheme has disappeared, and issue of positivity for the diffusion problem (although unresolved on distorted mesh and/or with varying cross-section) has been in some sense improved.