For diagonal conics, such as your example of $x^2+y^2-3z^2=0$, a non-zero rational solution exists *if and only if* one exists modulo all powers of all prime divisors of the coefficients and modulo powers of 2. And the powers you need to consider can also be bounded a priori. This is the Hasse-Minkowski theorem. Moreover, for those odd $p$ that don't divide any of the coefficients, a solution modulo $p$ always exists. So in your example, 2 and 3 are the only primes worth trying. All this can be found in Cassels, Lectures on Elliptic Curves, chapters 3,4,5. This works more generally for quadratic forms over number fields.


On the other hand for cubic forms, this local-global principle fails, so there may be no modulus that will exclude the existence of rational solutions, even when there really are none. A famous example due to Selmer is $3x^3+4y^3+5z^3=0$. This has solutions modulo all prime powers and over $\mathbb{R}$, but not over $\mathbb{Q}$. You can read more about this in Silverman, Arithmetic of Elliptic Curves.