= Solution
Assume the bounded set $\Omega$ is measurable, as needed to define its indicator in $L^q$. Put $w_j=f_j-f$ and choose a smooth <cutoff function> $\eta$ of compact support equal to one near $\Omega$. <Weak convergence> in the <H1 space> gives a uniform $H^1$ bound, by the <Uniform boundedness principle>, and therefore $u_j=\eta w_j$ is uniformly bounded in $H^1$ with fixed compact support.
Here is a direct Fourier proof of the local compactness in the <Rellich-Kondrachov compactness theorem>. For each frequency $\xi$, $\widehat u_j(\xi)\to0$, since integrating $w_j$ against the fixed compactly supported function $\eta(x)e^{-ix\cdot\xi}$ is a continuous linear functional on $H^1$. Also $|\widehat u_j(\xi)|\leq C$ uniformly, by <Cauchy-Schwarz inequality> on the fixed support. The <dominated convergence theorem> gives, for every fixed $R$,
$$
\int_{|\xi|\leq R}|\widehat u_j(\xi)|^2\,d\xi\longrightarrow0.
$$
The uniform derivative bound and the unitary <Plancherel theorem> control high frequencies:
$$
\int_{|\xi|>R}|\widehat u_j(\xi)|^2\,d\xi
\leq R^{-2}\int|\xi|^2|\widehat u_j(\xi)|^2\,d\xi
=R^{-2}\|\nabla u_j\|_2^2\leq C R^{-2}.
$$
First let $j\to\infty$, then $R\to\infty$. This proves $\|u_j\|_2\to0$, and hence $\|\chi_\Omega w_j\|_2\to0$.
Let $2^*=2n/(n-2)$. The <Sobolev embedding theorem> bounds $\|u_j\|_{2^*}$ uniformly. For $2<q<2^*$, choose $\theta\in(0,1)$ with $1/q=\theta/2+(1-\theta)/2^*$. <Interpolation of Lp norms> gives $\|u_j\|_q\leq\|u_j\|_2^\theta\|u_j\|_{2^*}^{1-\theta}\to0$. For $1\leq q<2$, <Holder inequality> on the fixed finite-measure support gives the same conclusion from $L^2$ convergence. Thus
$$
\boxed{\chi_\Omega f_j\longrightarrow\chi_\Omega f
\quad\text{strongly in }L^q,\qquad1\leq q<\frac{2n}{n-2}.}
$$
The same integral estimate covers $0<q<1$ if those spaces are interpreted with their usual quasi-norm. No conclusion at the critical exponent is asserted.
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