"We are not relations" does sound a little strange to me. But it also comes off as possibly a Southern saying (American South). And it fits since Forrest Gump was raised in Greenbow, Alabama, a southern state.
Looking at the definition of relations
a (1) : a person connected by consanguinity or affinity : relative (2) : a person legally entitled to a share of the property of an intestate
b : relationship by consanguinity or affinity : kinship
we see that "We are not relations" serves to mean that they are not family, not twins. So I don't believe it is a matter of correct or incorrect. It might be more appropriate or familiar to say if you are in certain regions.
Possible alternatives include
- We are not related.
- We are not relatives.
Including "we are not relations", they all mean "we are not family". I think I would've used 1. I think most people are familiar with these two.
Here is some ngram data on the matter.