Are there infinitely many natural numbers that can be represented as $\frac{a^2+b^2}{c^2+d^2}$ with coprime $a$, $b$, $c$, $d$? 
Are there infinitely many natural numbers that can be represented as the ratio
$$
\frac{a^2 + b^2}{c^2+d^2}
$$
where $a$, $b$, $c$, $d$ are coprime integers?

Is there a way to characterize all such numbers?

Excuse my late reaction, but I think the answer is already contained in the so called "Fermat Xmas theorem", which states that an odd prime $p$ is the sum of two (integral) squares iff $p \equiv 1$ mod $4$ (*). A consequence (due to Euler ?) is that a strictly positive integer is a sum of two squares iff in its prime decomposition, every factor   $p\equiv 3$ intervenes with an even exponent. Consider then an integer $N$ equal to a fraction whose numerator and denominator are both sums of two squares. The integrality of $N$ implies that each prime divisor of the denominator divides also the numerator, and with a lesser exponent. So that finally $N$ is already a sum of two squares.
(*) There exist quite a few different proofs of this Xmas theorem, but the record of brievity belongs to Don Zagier:« A one-sentence proof that every prime $p\equiv 1$(mod 4) is a sum of two squares », The American Mathematical Monthly, vol. 92, no 2,‎ 1990, p. 144.
