  We investigate the numerical stability of accelerated AdS black holes against
linear scalar perturbations. In particular, we study the evolution of a probe
non-minimally coupled scalar field on Schwarzschild and Reissner-Nordstr\"om
AdS black holes with small accelerations by computing the quasinormal modes of
the perturbation spectrum. We decompose the scalar field Klein-Gordon equation
and study the eigenvalue problem for its angular and radial-temporal parts
using different numerical methods. The angular part is written in terms of the
Heun solution and expanded through the Frobenius method which turns out to give
eigenvalues qualitatively similar to the ones obtained through the spherical
harmonics representation. The radial-temporal evolution renders a stable field
profile which is decomposed in terms of damped and purely imaginary
oscillations of the quasinormal modes. We calculate the respective frequencies
for different spacetime parameters showing the existence of a fine-structure in
the modes, for both real and imaginary parts, which is not present in the
non-accelerated AdS black holes. Our results indicate that the Schwarzschild
and Reissner-Nordstr\"om AdS black holes with small accelerations are stable
against linear scalar perturbations.
