peano arithmetic - finite or infinite many quadratic fields embedding into quaternion algebras?

To supplement David's answer: there is a standard "local-global" criterion for determining whether a quadratic field $K/mathbb{Q}$ can be embedded in a rational quaternion algebra $B/mathbb{Q}$.

For this recall that $B$ is said to be ramified at a prime number $p$ if $B_p = B otimes_{mathbb{Q}} mathbb{Q}_p$ is a division algebra. Moreover, we say that $B$ ramifies "at infinity" if $B_{infty} = B otimes_{mathbb{Q}} mathbb{R}$ is a division algebra.

It is known that a rational quaternion algebra $B$ is determined up to isomorphism by the set of ramified places $p leq infty$, that this set of places is finite and of even cardinality, and conversely for any finite set of even cardinality there is a rational quaternion algebra ramifying at these places.

Now, let $B$ be a rational quaternion algebra, $K$ a quadratic field, and $p leq infty$ a ramified place of $B$. Suppose that we have an embedding $K hookrightarrow B$. Then tensoring with $mathbb{Q}_p$ we get $K_p := K otimes mathbb{Q}_p hookrightarrow B_p$.

Now, if $p$ is inert or ramified in $K$, then $K_p$ is a quadratic field extension of $mathbb{Q}_p$, and it turns out that every such quadratic extension does indeed embed in $B_p$. However, if $p$ is split in $K$, then $K_p cong mathbb{Q}_p times mathbb{Q}_p$ has nontrivial idempotent elements, so cannot embed in $B_p$ if the latter is a division algebra. (If $p = infty$, then we say that $p$ is split in $K$ iff $K$ is a real quadratic field.)

In summary, this gives a necessary local criterion for the embeddability of $K$ into $B$: each ramified prime $p leq infty$ of $B$ is nonsplit in $K$. By the local-global theory of quaternion algebras over $mathbb{Q}$, it turns out that this necessary condition is also sufficient. In particular, the quadratic fields which embed into a given quaternion algebra are precisely those which are determined by finitely many splitting conditions.

It follows easily from this that there are infinitely many quadratic fields which embed in $B$, for instance any imaginary quadratic field $mathbb{Q}(sqrt{D})$ where $D$ is divisible by each finite ramified prime $p$ of $B$. Moreover, one can see that the set of such quadratic fields has, in some natural sense, positive density (as does its complement, unless $B cong M_2(mathbb{Q})$ in which case we recover the result that every quadratic field embeds, as one sees much more easily by a Cayley's Theorem / regular representation style argument).