We present a Coq formalization of the Quantified Reflection Calculus with one modality, or QRC₁. This is a decidable, strictly positive, and quantified modal logic previously studied for its applications in proof theory. The highlights are a deep embedding of QRC₁ in the Coq proof assistant, a mechanization of the notion of Kripke model with varying domains and a formalization of the soundness theorem. We focus on the design decisions inherent to the formalization and the insights that led to new and simplified proofs.