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Unfortunately not! You cannot infer globally generation of the vector bundle from globally generation of the tautological line bundle on the projectivized bundle. Ernesto Mistretta made me remark this once!
An aside comment: your globally generated condition implies that $\mathcal O_{\mathbb P(\Omega^1)}(1)$ is nef. Then, the second Segre number positive gives you bigness of $\mathcal O_{\mathbb P(\Omega^1)}(1)$. So you are looking at surfaces with big&nef cotangent sheaf, right?
