Is there a word for someone born through incest?
Solution 1:
"Inbred" is pretty close, though not strictly confined to incest, though in common usage it is meant as an insult and the incestuous implication is the primary intention (e.g. "inbred hick")
"Inbreeding" is the interbreeding of closely related individuals.
Solution 2:
I doubt there's a specific word - incestuous love-child is probably the closest you'll get.