We study arithmetic varieties $V$ attached to certain inner forms of
$\boldsymbol{Q}$-rank one of the split symplectic $\boldsymbol{Q}$-group of
degree two. These naturally arise as unitary groups of a 2-dimensional
non-degenerate Hermitian space over an indefinite rational quaternion division
algebra. First, we analyze the canonical mixed Hodge structure on the cohomology
of these quasi-projective varieties and determine the successive quotients of
the corresponding weight filtration. Second, by interpreting the cohomology
groups within the framework of the theory of automorphic forms, we determine the
internal structure of the cohomology “at infinity” of $V$,
that is, the part which is spanned by regular values of suitable Eisenstein
series or residues of such. In conclusion, we discuss some relations between the
mixed Hodge structure and the so called Eisenstein cohomology. For example, we
show that the Eisenstein cohomology in degree two consists of algebraic
cycles.