If some two vertices $u$, $v$ are not joined by a walk of length at most $2$ in $G$, then they are distinct, non-adjacent and have no common neighbor. In Ḡ they are adjacent, and every other vertex is adjacent to $u$ or to $v$. So any two vertices are joined in Ḡ by a walk that goes to $u$ or $v$, possibly crosses the edge $uv$, and leaves again, of length at most $3$.