Is it correct to say AX A= {(a, a) : a belongs to A}?

No.

A X A= { <x,y>: x e A & y e A}

Explanation: A X B = { <x,y>: x e A & y e B}

In other words, A X B is the class of all ordered pairs <x,y> such that x belongs to A and b belongs to B.

Therefore, A X A= { <x,y>: x e A & y e A}

What you wrote is syntactically ill-formed, and it’s suggested meaning is false, since no set contains every ordered pair consisting of its own members.