  We study quaternionic stochastic areas processes associated with Brownian
motions on the quaternionic rank-one symmetric spaces $\mathbb{H}H^n$ and
$\mathbb{H}P^n$. The characteristic functions of fixed-time marginals of these
processes are computed and allows for the explicit description of their
corresponding large-time limits. We also obtain exact formulas for the
semigroup densities of the stochastic area processes using a Doob transform in
the former case and the semigroup density of the circular Jacobi process in the
latter. For $\mathbb{H}H^n$, the geometry of the quaternionic anti-de Sitter
fibration plays a central role , whereas for $\mathbb{H}P^n$, this role is
played by the quaternionic Hopf fibration.
