Topological defects are ubiquitous on surfaces with orientational order fields. Here, we study equilibrium states generated by the feedback between geometry and nematic order on fluid membranes with an integer topological defect. When the Frank elastic constants associated with the orientational field dominate, the surfaces spontaneously deform toward an conical shape featuring an aster topological defect at its apex. In the case of vanishing tension this is a solution to the normal force balance. We show that the stability of the surface depends on the balance of the elastic parameters and the phase of the defect. When boundary constraints are introduced, we observe three distinct modes of deformation. These deformation modes take advantage of the way in which splay, twist and bend distortions of the director field can be exchanged on a curved surface. We discuss how these deformation modes are distinguished by their response to the cost of twist distortions and the existence of inverted solutions. Our findings show that fusion of +1/2 topological defect pairs can reduce the total energy of deformable surfaces. Finally, we argue how these results can be relevant for biological systems.