  We study the constraint equations for a class of scalar-tensor effective
field theories of gravity, including the operators up to $4$ derivatives in the
action ($4\partial$ST). We extend the conformal transverse traceless and
conformal thin sandwich methods of General Relativity to rewrite the constraint
equations of the scalar-tensor theory as a set of elliptic partial differential
equations. It is shown that, at weak coupling, a unique solution exists to the
corresponding elliptic boundary value problems on asymptotically Euclidean
initial slices under similar conditions as in the case of General Relativity.
Furthermore, we find a generalization of the Bowen-York solution for
$4\partial$ST theories, too. These results demonstrate that standard methods
for constructing initial data in General Relativity are applicable (with
minimal modification) to weakly coupled $4\partial$ST theories.
