Suppose $D$ is an indefinite quaternion (division) algebra over $\mathbb{Q}$. For $a,b\in D$, we say $a$ and $b$ are conjugate if and only if there exists an element $d\in D$ such that $a=dbd^{-1}$.
My question is whether $|D/\sim|$ is finite or not?
The center of $D$ is isomorphic to $\mathbb{Q}$, and each element is its own conjugacy class. So you at the very least have a conjugacy class for each element of $\mathbb{Q}$, in particular there is an infinite number of them.
More generally, take any non-zero $x\in D$. Then the only elements in the line $\mathbb{Q}x$ that are conjugate to $x$ are $x$ and (possibly) $-x$. This is easy to see: any conjugate of $x$ has the same norm as $x$, but $N(\lambda x)=\lambda^2N(x)$ so the only possibility is $\lambda=\pm 1$.
This means that every line in $D$ intersects an infinite number of conjugacy classes.