How do I construct the score process of a Markov model and verify that it is a Martingale?
The derivation of the score process is correct. To verify that the process is a Martingale, recall the definition. It becomes clear that if we substitute $W_{t+1}$ back into the equation
$$
s_t(\theta \mid \textbf X) =
\begin{bmatrix}
 (1 - \beta_0) \sum_{j=1}^t W_j \\
 \sum_{j=1}^t W_j (X_{j-1} - \alpha_0)
\end{bmatrix}.
$$
Because $W_{t+1}$ are Normal with mean 0 and variance 1 (I'm assuming they're iid), then
$$
E[s_{t+1} \mid s_t ] = s_t + E \left [
\begin{matrix}
(1 - \beta_0) W_{t+1} \\
W_{t+1} (X_t - \alpha_0)
\end{matrix} \middle |
s_t
\right ]
= s_t
$$ 
and we are finished.