= Solution
For a <continuous semimartingale> $X$, its <quadratic variation> is the continuous increasing zero-starting process obtained as the limit, uniformly on compact time intervals in probability, of sums of squared increments along deterministic partitions whose mesh tends to zero. If $X=X_0+N+V$ is its continuous local-martingale/finite-variation decomposition, then $[X]=[N]$; finite-variation terms and their cross sums vanish.
Write $D=d\mathbb Q/d\mathbb P$. If $\mathbb P(E_n)\to0$, then for every $R>0$,
$$
\mathbb Q(E_n)\leq R\mathbb P(E_n)+\mathbb E_{\mathbb P}[D\mathbf1_{\{D>R\}}].
$$
First send $n$ to infinity and then $R$ to infinity. Thus convergence in probability under $\mathbb P$ implies convergence in probability under $\mathbb Q$, also for a supremum on a compact time interval. The same squared-increment sums converge under $\mathbb Q$ to the $\mathbb P$ version of $[X]$. Since $X$ is also a <semimartingale> under $\mathbb Q$, these sums converge to its $\mathbb Q$ <quadratic variation>. Uniqueness of limits in probability gives equality at every fixed time; continuity and countably many rational times give
$$
\boxed{[X]^{\mathbb Q}=[X]^{\mathbb P}\quad\mathbb Q\text{-indistinguishably}.}
$$
Absolute continuity suffices; equivalence is not required. This is <quadratic variation under an absolutely continuous measure change>.
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