The paper Eisenstein series and quantum affine algebras by Kapranov makes contact between automorphic forms and quantum groups. I haven't found even one other paper devoted to this theme.
Have other authors come at this, perhaps from other perspectives?
There are at least two research strands that fit the description - if by automorphic forms you allow me to consider the function field versions. The one closest to your question is in the direct line of Kapranov's very influential paper. The topic of Hall algebras is extremely active (I recommend Schiffmann's beautiful survey). Since Ringel and Lusztig, Hall algebras have been one of the primary means to understand quantum groups. On the other hand, since Kapranov's paper in particular there's been a realization that the entire geometric Langlands program for general linear groups is a subset of the study of the [categorified] theory of Hall algebras - namely the case of Hall algebras for categories of coherent sheaves on curves. The Hecke operators on functions on moduli spaces of bundles are a part of the multiplication action of the Hall algebra of this category on itself, and if we replace functions with sheaves (study Hall categories) we recover the basic questions in the function field version of the theory of automorphic forms. There are innumerable papers this relates to, but I think the most directly relevant to your question are the wonderful papers of Schiffmann and Vasserot on Hall algebras, Macdonald polynomials, geometric Eisenstein series and geometric Langlands (see arXiv search for those two names).
The other major direction is the "quantum geometric Langlands correspondence", which is a q-deformation of the [categorified] theory of automorphic forms on function fields (ie geometric Langlands), which is directly related to the representation theory of quantum groups. (There are various close connections between undeformed, plain old geometric Langlands and quantum groups as well, see e.g. Arkhipov-Bezrukavnikov-Ginzburg and other [amazing] papers of Bezrukavnikov, but this is in a somewhat different direction, though of course everything is related.) Unfortunately not that much is written about quantum geometric Langlands -- its origins are in works of Feigin-Frenkel and Beilinson-Drinfeld, and the idea has been around since the late 90s, but it's hard to find in the literature until recently (though see the withdrawn preprint by Stoyanovsky, still available on arXiv in early versions, for the general idea). In any case a major paper on this topic is Gaitsgory's paper constructing quantum groups directly out of a deformed version of the geometric Satake correspondence. This is the first step in a program of Gaitsgory and Lurie on quantum geometric Langlands which is not publicly documented AFAIK. A recent paper on the topic is Travkin. The subject got a major push from the fact that it arises very naturally from the gauge theory point of view on geometric Langlands due to Kapustin-Witten. But maybe this response is not the right place to survey what this conjecture is actually about..
