In this paper, we calculate the expected values of the first and second Zagreb indices, denoted as $\textbf{E}\left(M_1\right)$ and $\textbf{E}\left(M_2\right)$ respectively, as well as the expected value of the forgotten index, $\textbf{E}\left(F\right)$, for two models of random bipartite graphs. To evaluate our findings, we establish the growth rate by demonstrating that for a random bipartite graph $G$ of order $n$ in either model, the expected value of $M_1(G)$ is $O\left( n^3 \right)$. Furthermore, we prove that the expected values of $M_2(G)$ and $F(G)$ are both $O\left( n^4 \right)$.