  We investigate the dynamical behavior of a scalar field non-minimally coupled
to Einstein's tensor and Ricci scalar in geometries of asymptotically de Sitter
spacetimes. We show that the quasinormal modes remain unaffected if the scalar
field is massless and the black hole is electrically chargeless. In the massive
case, the coupling of both parameters produces a region of instability in the
spacetime determined by the geometry and field parameters. In the Schwarzschild
case, every solution for the equations of motion with $\ell>0$ has a range of
values of the coupling constant that produces unstable modes. The case $\ell=0$
is the most unstable one, with a threshold value for stability in the coupling.
For the charged black hole, the existence of a range of instability in $\eta$
is strongly related to geometry parameters presenting a region of stability
independent of the chosen parameter.
