  We classify curvature-adapted real hypersurfaces $M$ of non-flat quaternionic
space forms $\mathbb HP^m$ and $\mathbb HH^m$ that are of Chen type 2 in an
appropriately defined (pseudo) Euclidean space of quaternion-Hermitian
matrices, where in the hyperbolic case we assume additionally that the
hypersurace has constant principal curvatures. In the quaternionic projective
space they include geodesic hyperspheres of arbitrary radius $r \in (0, \pi/2)$
except one, two series of tubes about canonically embedded quaternionic
projective spaces of lower dimensions and two particular tubes about a
canonically embedded $\mathbb CP^m \subset \mathbb HP^m $. On the other hand,
the list of 2-type curvature-adapted hypersurfaces with constant principal
curvatures in $\mathbb HH^m$ is reduced to geodesic spheres and tubes of
arbitrary radius about totally geodesic quaternionic hyperplane $\mathbb
HH^{m-1}.$ Among these hypersurfaces we determine those that are mass-symmetric
or minimal. We also show that the horosphere $H_3$ in $\mathbb HH^m $ is not of
finite type but satisfies $\Delta^2\widetilde x =$ const.
