When you're starting off, avoid using Print or Do. Print just causes something to appear on the screen. You don't want to use print. You almost never want to actually use Print in Mathematica. You want to return the actual values so that you can use them for later. Instead of these, let's use Table:
Let's break your code down into some simpler pieces. This will make it easier to maintain and understand. Here's a function that generates a list of the values you are interested in:
generateValues[m_, n_] := {m^2 - n^2, 2 m n, m^2 + n^2, m^2 - n^2 - 2 m n};
Table[generateValues[m, n], {n, 1, 5}, {m, n + 1, 5}]
Now let's make a function that returns True only when the first three elements of a list satisfy a^2+b^2=c^2.
isItPythagorean[list_] := (list[[1]]^2 + list[[2]]^2 == list[[3]]^2)
Let's test that it works:
isItPythagorean[{3, 4, 5}]
True
We can apply it inside the Table:
Table[isItPythagorean[generateValues[m, n]], {n, 1, 5}, {m, n + 1, 5}]
{{True, True, True, True}, {True, True, True}, {True, True}, {True}, {}}
