Is the cardinality of the rationals the same on the unit interval? Let $\mathscr{G}$ be a compact subset of $\mathbb{R}$. Denote by $\mathbb{Q} ( \mathscr{G})$ the set of all rational numbers in $\mathscr{G}$. Does $| \mathbb{Q} ( \mathscr{G}_1)| = |\mathbb{Q} (\mathscr{G}_2)|$ for $\mu(\mathscr{G}_1) > \mu(\mathscr{G}_2)$ where $\mu$ is the Lebesgue Measure?   
i.e. are there more rationals in $[0,2]$ than $[0,1]$?

No, there are not more rationals in $[0,2]$ than in $[0,1]$.  In fact, there are not even more elements in $[0,2]$ and $[0,1]$, despite the fact that these two sets have different measure, since there exists a bijection between them.
Note that measure and cardinality behave, in some ways, "independently" of one another.  Another noteworthy example is that of the Cantor set and the rationals.  We note that although the Cantor set has cardinality greater than that of the rationals, both sets have measure $0$.
Also, note that we can have (measurable) sets of any size that contain no (or any finite number of) rational numbers.  For example, $\mathscr{G} = [0,2]\setminus\mathbb{Q}$ satisfies $\mu(\mathscr{G}) = 2$ while $|\mathbb{Q}(\mathscr{G})| = 0$.
