I came across this notation while at this MathOverflow thread and I could not find its meaning. It makes the biggest sense that $f(x,y) \in \Bbb Q[x,y] $ represents any continuous function on the interval $[x,y] \in \Bbb Q$ and thus $\Bbb Q[x,y]$ is a set of such functions. However, I am not sure.
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It means a polynomial in two variables with coefficients in the rational numbers $\mathbb Q.$
Or, it means the rationals adjoined with the elements x,y as a ring. So in @Will Jagy 's answer, x,y are indeterminants and therefore transcendentals. But x,y could also be other numbers that are not transcendental (i.e. algebraic), but you would need explicit values. For example, $x=\sqrt2$ and $y=\sqrt3$ would also work, but these are algebraic (i.e. solutions of polynomials over the rationals). This matters because it results in very different rings.