If were told that something is gamma distributed with mean $m$, are we able to use that to find one of the parameters? Like if were told that something is exponentially distributed with mean $m$, then the it's exponentially distributed with parameter $\lambda =\frac1m$. Can we also do this with the gamma? (not necessarily put the mean in the denominator, but use this info to our advantage).
No, because the mean is a function of both parameters. The best you can do is use $m$ to constrain the relation between the parameters, i.e. $m\beta=\alpha$. To determine one (and therefore both) of the parameters you'll need another measure, for example the variance.