Reference: "The Arithmetic of Elliptic Curves" (Silverman, second edition), Exercise 1.11. (b).
Consider two circles $X^2 + Y^2 = p \cdot Z^2$ and $X^2 + Y^2 = q \cdot Z^2$.
If $p$ and $q$ are distinct primes congruent to $1$ modulo $4$, then the circles are isomorphic over $\mathbb{Q}$. I understand why (essentially because $p$ and $q$ are the sum of two squares, such that both circles are isomorphic to the projective line over $\mathbb{Q}$).
If $p$ and $q$ are distinct primes congruent to $3$ modulo $4$, then the circles are not isomorphic over $\mathbb{Q}$. According to Silverman's exercise, it is true, but I do not understand why (but I understand that there is no rational point on such circles).
If $p = 1\ (mod\ 4)$ and $q = 3\ (mod\ 4)$, I do not know whether the circles are isomorphic or not over $\mathbb{Q}$ (perhaps depending upon a further condition).