Possibly, but it would join other false conjectures such as Euler's sum of powers conjecture - posed in 1769 and no counterexample found until 1966. There's only been three primitive counterexamples found so far.
Not even the same implications. All empirical evidence strongly support the Goldbach conjecture. Any counterexample would mean an entire field of Mathematics has to be rewritten.
Many natural questions would arise, starting with βIs this the only counterexample?β