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Let ${\Sigma }^2$ be an immersed surface in $M^2\left(c\right)\times ℝ$ with constant mean curvature. We consider the traceless Weingarten operator $\phi$ associated to the second fundamental form of the surface, and we introduce a tensor $S$, related to the Abresch-Rosenberg quadratic differential form. We establish equations of Simons type for both $\phi$ and $S$. By using these equations, we characterize some immersions for which $\left|\phi \right|$ or $\left|S\right|$ is appropriately bounded.