  We consider intertwining relations of the augmented $q$-Onsager algebra
introduced by Ito and Terwilliger, and obtain generic (diagonal) boundary
$K$-operators in terms of the Cartan element of $U_{q}(sl_2)$. These
$K$-operators solve reflection equations. Taking appropriate limits of these
$K$-operators in Verma modules, we derive $K$-operators for Baxter Q-operators
and corresponding reflection equations.
