A Maxwell-Boltzmann distribution, written as a Gamma distribution

Just compare two formulae: $$ {\displaystyle f(x) ={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{\alpha -1}e^{-\beta x}},\qquad f(E)=2{\sqrt {{\frac {E}{\pi }}}}\left({\frac {1}{kT}}\right)^{{3/2}}\exp \left({\frac {-E}{kT}}\right). $$

We can see that $\beta=\frac1{kT}$ and $\alpha=\frac32$.

Your confusion may arise from the fact that besides parameters $\alpha,\beta$ (shape, rate) for gamma distribution, parameters $\alpha,\theta$ (shape,scale) are also widely used ($\theta=1/\beta$). What adds to the confusion, is that scale parameter is often denoted as $\beta$ as well. So seeing just the value of $\beta$ one can never be sure what it really means and needs to refer to documentation. In particular, the function std::gamma_distribution used in your link has the exact this issue (although they admit that the notation is not universal):

In other words parameters to put into std::gamma_distribution should be $\alpha=3/2$ and $\beta=kT$ ($\theta=kT$).