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In this paper, we present the preliminaries of <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-calculus for functions of two variables. Furthermore, we prove some new Hermite-Hadamard integral-type inequalities for convex functions on coordinates over <inline-formula> <math display="inline"> <semantics> <mrow> <mo>[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>]</mo> <mo>&#215;</mo> <mo>[</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo>]</mo> </mrow> </semantics> </math> </inline-formula> by using the <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-calculus of the functions of two variables. Furthermore, we establish an identity for the right-hand side of the Hermite-Hadamard-type inequalities on coordinates that is proven by using the <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-calculus of the functions of two variables. Finally, we use the new identity to prove some trapezoidal-type inequalities with the assumptions of convexity and quasi-convexity on coordinates of the absolute values of the partial derivatives defined in the <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-calculus of the functions of two variables.