For anyone still struggling with Koops terrible notation: The problem is that Koop uses neither the scale nor the rate parametrization, but rather a "mean,degrees of freedom" parametrization (see Appendix, Def. B. 22).
The distribution of $h$ in a proper parametrization (shape, rate) is thus
$$
h \sim \text{Gamma}(shape = \underline{\nu}/2 , rate = \underline{\nu s}^2 / 2)
$$
using Koops notation for the parameters.
Go to the same site on the following sub-page:
https://onlinecourses.science.psu.edu/stat414/node/278
You will see more clearly that they specify the simple linear regression model with the regressor centered on its sample mean. And this explains why they subsequently say that $\hat \alpha$ and $\hat \beta$ are independent.
For the case when the coefficients are estimated with a regressor that is not centered, their covariance is
$$\text{Cov}(\hat \alpha,\hat \beta) = -\sigma^2(\bar x/S_{xx}), \;\;S_{xx} = \sum (x_i^2-\bar x^2) $$
So you see that if we use a regressor centered on $\bar x$, call it $\tilde x$, the above covariance expression will use the sample mean of the centered regressor, $\tilde {\bar x}$, which will be zero, and so it, too, will be zero, and the coefficient estimators will be independent.
This post, contains more on simple linear regression OLS algebra.
Best Answer
The $\hat \alpha $ and $\hat \beta$ commonly seen in regression equations are parameters that are estimated. They indeed represent numbers, but we don't know their value. Once they are estimated the estimated numbers can be substituted in their place.
The distribution of the parameters comes from the fact that there is uncertainty in the estimation process (due to possible noise and random sampling). If a different set of samples was drawn from the population, the estimates would vary. The distribution of the parameter estimates is generally assumed to be normal, with the estimate as the mean, and variance that is generally rather easy to estimate as well, based on the residuals.