  The set of random walks with different step sets (of short steps) in the
quarter plane has provided a rich set of models that have profoundly different
integrability properties. In particular, 23 of the 79 effectively different
models can be shown to have generating functions that are algebraic or
differentiably finite. Here we investigate how this integrability may change in
those 23 models where in addition to length one also counts the number of sites
of the walk touching either the horizontal and/or vertical boundaries of the
quarter plane. This is equivalent to introducing interactions with those
boundaries in a statistical mechanical context. We are able to solve for the
generating function in a number of cases. For example, when counting the total
number of boundary sites without differentiating whether they are horizontal or
vertical, we can solve the generating function of a generalised Kreweras model.
However, in many instances we are not able to solve as the kernel methodology
seems to break down when including counts with the boundaries.
