This article develops a factorization-based operator framework for two-variable discrete Gould–Hopper–Appell polynomials within Δδ-calculus. Following the Infeld–Hull factorization principle, the framework employs degree-dependent sequences of operators, rather than a single pair of ordinary quasi-monomial operators. From the generating function, we derive the recurrence relation, lowering and raising operators, and normalized integro-type operators. Their successive degree-matched compositions yield explicit difference, integro-difference, and partial-difference equations. Exact triangular Volterra integral representations are also established through the associated differential representation. The general results are specialized to Gould–Hopper–Bernoulli, Gould–Hopper–Euler, and Gould–Hopper–Genocchi families, with the Genocchi case treated separately. Numerical investigations of complex zeros, polynomial surfaces, generating-function reconstruction, and quantitative convergence as δ→ 0 complement the analytical results and illustrate the discrete-to-continuous behavior of the proposed polynomial families.