We study the fully discrete elliptic integrable model Q4 and its immediate trigonometric and rational counterparts (Q3, Q2, and Q1). Singular boundary problems for these equations are systematized in the framework of global singularity analysis. We introduce a technique to obtain solutions of such problems, in particular, constructing the exact solution on a regular singularity-bounded strip. The solution technique is based on the multidimensional consistency and uses new insights into these equations related to the singularity structure in multidimensions and the identification of an associated tau-function. The obtained special solutions can be identified with open boundary problems of the associated Toda-type systems, and have interesting application to the construction of periodic solutions.