Betweenness centrality is defined on connected graphs. Suppose a graph $G$ with two connected components, $G_1$ and $G_2$, and consider a node $v \in G_1$, then the betweenness centrality for $v$ is
$$C_B(u)=\sum_{s,t \in G}{\frac{\sigma_{st}(v)}{\sigma_{st}}}=\sum_{s,t \in G_1}{\frac{\sigma_{st}(v)}{\sigma_{st}}}+\sum_{s,t \in G_2}{\frac{\sigma_{st}(v)}{\sigma_{st}}}+\sum_{s \in G_1,t \in G_2}{\frac{\sigma_{st}(v)}{\sigma_{st}}}=$$$$C_B(u)=\sum_{s,t \in G}{\frac{\sigma_{st}(v)}{\sigma_{st}}}=\sum_{s,t \in G_1}{\frac{\sigma_{st}(v)}{\sigma_{st}}}+\sum_{s,t \in G_2}{\frac{\sigma_{st}(v)}{\sigma_{st}}}+2\sum_{s \in G_1,t \in G_2}{\frac{\sigma_{st}(v)}{\sigma_{st}}}=$$
$$\sum_{s,t \in G_1}{\frac{\sigma_{st}(v)}{\sigma_{st}}}+\sum_{s,t \in G_2}{\frac{\sigma_{st}(v)}{\sigma_{st}}}+\sum_{s \in G_1,t \in G_2}{\frac{\sigma_{st}(v)}{0}}$$$$\sum_{s,t \in G_1}{\frac{\sigma_{st}(v)}{\sigma_{st}}}+\sum_{s,t \in G_2}{\frac{\sigma_{st}(v)}{\sigma_{st}}}+2\sum_{s \in G_1,t \in G_2}{\frac{0}{0}}$$
But if you want to treat each connected component as a single graph, you can also calculate the betweenness centrality independently for each component.