The error is in \mbox{\sqrt[]{2}}, because \mbox switches to text mode.
However, this seems a consequence of another mistake: for implementation reasons, an empty optional argument to \sqrt causes the symbol to be misplaced. Avoid \sqrt[]{2} and use the correct \sqrt{2}.
\documentclass{article}
\usepackage{amsmath,amssymb}
\newcommand{\BQ}{\mathbb{Q}} % <--- with braces
\begin{document}
\begin{tabular}{l l}
Wrong: & $\forall a, b \in \BQ, x = a + b\sqrt[]{2}$ \\
Better: & $\forall a, b \in \BQ, x = a + b\sqrt{2}$ \\
Best: & for all $a, b \in \BQ$, $ x = a + b\sqrt{2}$
\end{tabular}
\medskip
Cube roots don't have the problem: $a+b\sqrt[3]{2}$
\end{document}

I prefer not to use \forall, but in any case, it should be two formulas: $\forall a,b\in\BQ$, $x=a+b\sqrt{2}$.
Note also that the correct syntax is \mathbb{Q} instead of \mathbb Q.
\BQcommand? – Jun 14 '17 at 06:12\sqrtin text mode (inside a box): it's not clear to me why ... – Joseph Wright Jun 14 '17 at 06:15\mboxto get somethingnicerin this case; secondly, you better ask your question via anMWE; thirdly, this code generates what you want, doesn't it?! `\documentclass{article}\usepackage{amssymb}
\begin{document}
$(\forall a, b \in \mathbb{Q}) x = a + b\sqrt{2}$
\end{document} `
– Jun 14 '17 at 06:20[], avoid it when no index is needed. So\sqrt{2}and\sqrt[3]{3}, but not\sqrt[]{2}. – egreg Jun 14 '17 at 09:00$$, like egreg did in his answer. I don't mean doing$x$ $\in$ $\mathbb{Q}$-- it is tottally wrong, but splitting sentence into logical parts will give you better spacing and less problems with linebreaking. – Michael Fraiman Jun 14 '17 at 09:28