What is a Q distribution? In this 1967 paper by Hogben it is shown that the sample correlation coefficient $r=\frac{\sum_{i=1}^n(Y_i-\bar Y)(x_i-\bar x)}{\sqrt{\sum(x_i-\bar x)^2\sum(Y_i-\bar Y)^2}}$ is Q-distributed with n-2 degrees of freedom and noncentrality parameter $\theta=\beta/\sigma\sqrt{\sum(x_i-\bar x)^2}$ under the standard linear model assumption.
Roughly speaking $r$ can be written as $\frac{W}{\sqrt {W^2+X^2}}$ with $W\sim \mathcal N(\beta/\sigma\sqrt{\sum(x_i-\bar x)^2}, 1)$ and $X^2\sim \chi^2(n-2)$. It seems that the Q distribution is defined in Hogben et al. 1964[a] and [b] but I'm not able to find the papers.

If $W$ is a random normal variate with mean $\theta$ and variance 1, and $Z^2$  is independently distributed as Chi-squared with $n$ degrees of freedom, then the random variable $Q$ with non-centrality $\theta$ and $n$ degrees of freedom is defined by:
$$
Q = \frac{W}{\sqrt{W^2 + Z^2}}
$$
From:
Hogben, D., Pinkham, R.S. and Wilk, M.B., 1964. An Approximation to the Distribution of Q (A Variate Related to the Non-Central t) 1. The Annals of Mathematical Statistics, pp.315-318.
https://projecteuclid.org/download/pdf_1/euclid.aoms/1177703753
