Yes, Gru has a winning strategy. First Gru chooses a square region $S$ of the grid with side length $10^9$. Then, until all edges on the boundary of $S$ are painted, on each move Gru paints an arbitrary such unpainted edge. After no more than $4 \cdot 10^9$ moves by Gru, the boundary of $S$ must be painted completely. At this time, the minions may have painted at most $4 \cdot 2022 \cdot 10^9$ edges, which is much less than the number of edges inside $S$, so there is at least one unpainted edge inside $S$. Then, on each move, Gru paints an arbitrary unpainted edge inside $S$. At some point, there will be exactly one unpainted edge inside $S$. The two cells adjacent to this edge will form the subgrid that Gru needs to win.