Here our assumptions are that $\alpha$, $\beta$, and $\gamma$ are Gaussian Integers and ($\alpha$, $\beta$) = $\mathbb{G}$.
To prove this, I let $\gamma = \alpha\delta$ and $\gamma = \beta\phi,$ where $\delta, \phi\in\mathbb{G}$. This is from the definition of divisibility. I then considered $\gamma\gamma = \alpha\delta\beta\phi = (\alpha\beta)\rho,$ with $\rho = \delta\phi$. I.e., $\gamma\gamma|\alpha\beta$
I then showed that $\gamma|\gamma\gamma$, and via the transitive property $\gamma|\alpha\beta$.
However, my concern is that I did not use the assumption that ($\alpha, \beta$)=$\mathbb{G}$. I think that needs to be included, but I'm not sure how/where. Any help would be greatly appreciated!