In this work, we introduce the concept of double quantum integrals for
the interval-valued functions of two variables. We offer several new
inclusions of the Hermite-Hadamard type for co-ordinated convex
interval-valued functions using the newly defined integrals. Moreover,
we prove trapezoidal type inequalities for interval-valued functions of
two variables using the ideas of the Pompeiu--Hausdorff distance between
the intervals. It is also revealed that the results offered in this work
are the generalization of several existing results.