Let $B$ be an indefinite quaternion algebra over $\mathbb{Q}$ of discriminant $D$, and $\mathcal{O}_N$ be an Eichler order of level $N$. Is there an element $x\in \mathcal{O}_N$ such that its reduced norm $\mathrm{Nrd}(x)=-1$?

There exists an element $x\in B$ such that $\mathrm{Nrd}(x)=-1$. This follows from the norm theorem. But I don't know if the existence stays to be true if $B$ is replaced with $\mathcal{O}_N$.

The question is interesting in the following way. According to the paper of S. Molina, section 2.2, Shimura curves $X_0(D,N)$ (defined as the upper half plane quotient by the group of norm one elements in $\mathcal{O}_N$) parametrizes abelian surfaces with QM by $\mathcal{O}_N$. Fix an isomorphism $i: B\otimes \mathbb{R} \to \mathrm{Mat}_2(\mathbb{R})$. Given $z$ in the upper half plane, the $\Gamma_0(D,N)$-orbit of $z$ corresponds to the isomorphic class of pairs $(\mathbb{C}^2/\Lambda_z, i)$, where $\Lambda_z$ is the lattice $i(\mathcal{O}_N)\begin{bmatrix} z \\ 1\end{bmatrix}$.

**Question**: why all abelian surfaces with QM by $\mathcal{O}_N$ arises this way?

There is nothing special in taking $i(\mathcal{O}_N)\begin{bmatrix} z \\ 1\end{bmatrix}$. We can just take any invertible left-ideal $I$ of $\mathcal{O}_N$ and look at the lattice $i(I)\begin{bmatrix} z \\ 1\end{bmatrix}$. Since the class number is one, any such ideal is equal to $\mathcal{O}_Ny$ for some $y\in B$. So after multiplying a scalar, the lattice $i(I)\begin{bmatrix} z \\ 1\end{bmatrix}$ is equal to $i(\mathcal{O}_N)\begin{bmatrix} i(y)z \\ 1\end{bmatrix}$.

Everything works well if $\mathrm{Nrd}(y)>0$. However, in the case $\mathrm{Nrd}(y)<0$, then $i(y)z$ is no longer on the upper half plane. This can be remedied if there is an element $x\in \mathcal{O}_N$ such that $\mathrm{Nrd}{x}=-1$, in which case $i(\mathcal{O}_N)\begin{bmatrix} i(y)z \\ 1\end{bmatrix}$ coincides with $i(\mathcal{O}_N)\begin{bmatrix} i(xy)z \\ 1\end{bmatrix}$ up to a scalar. If such an element does not exists, I am afraid that we might have to throw in the lower half plane as well in defining the Shimura curve, and the quotient by $\Gamma_0(D,N)$ will not be connected.

In fact, in section IX.5 of "Introduction to Algebraic and Abelian Functions" by S. Lang, he explicitly assumed that there is an element in the order whose reduced norm is -1. Is there an theorem that says such an element will always exists in the indefinite case, and his assumption is just for exposition convenience rather than out of necessity?