Fn and m are integers and n2 m2 is even, which of the following is impossible?

So, I think that you mean that

n²+m² is even? (the "plus" sign is missing, but I think this is correct).

this can only happen if either:

both n and m are even
or
both n and m are odd

That's because a sum of two expressions is even if either both or none of the expressions summed is even, and a power of a number is even only if the number is even too.

are those some of the options?

yes!
n and m are even,- possible!
 n and m are odd, -possible!
n+m is even, - possible; it would be the case if either both n and m are even or both n and m are odd n+m is odd,

so the option we'll choose is:

none of these are impossible