In fact, without loss of generality, we can compute the probability of hitting the origin when we start at any \((n,m)\), where \(m \ge 0\). This can be treated by the inductive system: \[P^m_n = \begin{cases} \bar\gamma^n & \text{if } m = 0 \\ \sum_{k \in \mathbb{Z}} \frac{\bar\gamma^{|n-k|}}{\sqrt{5}} P^{m-1}_k & \text{otherwise} \end{cases}\]
This can probably yield a algebraic solution if we break the absolute sum into smaller piece, but for now it suffices to just be able to compute this for any \((n,m)\).