[Note: although this question has an accepted answer, the investigation is not finished yet. I encourage you to post your findings.]
Who first introduced the notation "$X \sim Q$", meaning that $Q$ is the probability distribution for $X$, and its related meanings?
Classical texts such as Jeffreys's Theory of Probability (3rd ed. 1961, 2nd ed. 1948) and Fisher's Statistical Methods for Research Workers (13th repr. ed. 1963) and Statistical Methods and Scientific Inference (1956) do not seem to use it (in Jeffreys "$\sim$" denotes the logical not).
I checked also the references given in this informative answer, but did not find anything relevant there.
(This notation is by no means universal in recent times or today, see eg Mosteller & Tukey's Data Analysis and Regression (1977) or Jaynes's Probability Theory (2003). In fact it does not make much sense when probability is seen as an extension of formal logic (eg Keynes, Johnson, Jeffreys, Pólya, Hailperin, Jaynes), defined over propositions rather than random variables.)
Edit (2021-06-24): The earliest reference I found so far is a paper by Khatri written in 1965 and published 1967, where the notation is introduced on p. 1854. The earliest textbook reference found so far is by Srivastava & Khatri, 1979, p. 41.
John Aldrich also kindly replied to my inquiry, saying that he never investigated the origin of this notation.