  Let $n$ be an integer such that the modular curve $X_0(n)$ is hyperelliptic
of genus $\ge2$ and such that the Jacobian of $X_0(n)$ has rank $0$ over
$\mathbb Q$. We determine all points of $X_0(n)$ defined over quadratic fields,
and we give a moduli interpretation of these points. As a consequence, we show
that up to $\overline{\mathbb Q}$-isomorphism, all but finitely many elliptic
curves with $n$-isogenies over quadratic fields are in fact $\mathbb Q$-curves,
and we list all exceptions. We also show that, again with finitely many
exceptions up to $\overline{\mathbb Q}$-isomorphism, every $\mathbb Q$-curve
$E$ over a quadratic field $K$ admitting an $n$-isogeny is $d$-isogenous, for
some $d\mid n$, to the twist of its Galois conjugate by some quadratic
extension $L$ of $K$; we determine $d$ and $L$ explicitly.
