  We study two-dimensional $s$-wave topological superconductivity with Rashba
spin-orbit coupling and Zeeman field in Penrose and Ammann-Beenker
quasicrystals. By solving the Bogoliubov-de Gennes equations self-consistently
for not only the superconducting order parameter, but also the spin-dependent
Hartree potential, we show the stable occurrence of TSC with broken
time-reversal symmetry in both Penrose and Ammann-Beenker quasicrystals. The
topological nature of the quasicrystalline system is signified by the Bott
index $B$. Topological phase transitions are found to occur, where $B$ changes
between 0 and $\pm 1$, as the chemical potential or Zeeman field is varied. In
terms of self-consistent solutions, we demonstrate the existence of a Majorana
zero mode per edge or vortex when $B=\pm 1$, consistently with the
bulk-edge/defect correspondence for periodic systems.
