Let $P\_1,\dots,P\_4$ be four projective plane conics without a common point. A dimension count tells us that there is a 9-dimensional space of quadratic relations among the defining equations $p\_j=0$ for $P\_j.$ Indeed, we have a $24=4\times 6$-dimensional vector space of coefficients that is mapped to the 15-dimensional space of plane quartics via the map $(q\_1,\dots,q\_4)\mapsto \sum\_{k=1}^4 q\_k p\_k$. 


Thus, apart from the 6 obvious Koszul relations $(p\_2,-p\_1,0,\dots,0)$, there should be 3 more.
Do they have a good description? (At least, in the generic case of no triple of $P\_j$ having a point in common, too?)



relations between 4 plane conics
Maybe this is something like what you have in mind:


Let $V$ be a (complex) vector space of dimension $3$. It is easy to show that a generic subspace $P\subset S^2(V^\*)$ of dimension $4$ can be written as 
$$
P = \mathrm{span}\{\ {x\_1}^2,\ {x\_2}^2,\ {x\_3}^2,\ x\_1x\_2{+}x\_2x\_3{+}x\_3x\_1\ \}
$$
for some basis $x\_1,x\_2,x\_3$ of $V^\*$. (There are only a finite number of other 'nongeneric' cases to be handled.)


It seems that you are asking for a 'natural' basis for the kernel of the multiplication map
$$
\mu:P\otimes S^2(V^\*)\longrightarrow S^4(V^\*)
$$
Now, $P\subset S^2(V^\*)$ is invariant under permutations of the basis $x\_1,x\_2,x\_3$, a group isomorphic to $S\_3$. The action of $S\_3$ on $S^2(V^\*)$ then has a natural complement to $P$, which is the space
$$
W = \mathrm{span}\{\ \ x\_1x\_2{-}x\_2x\_3,\ x\_2x\_3{-}x\_3x\_1\ \}.
$$
As representations of $S\_3$, $W$ is irreducible and $P \simeq \mathbb{C}\oplus\mathbb{C}\oplus W$. 


Now, we have the natural decomposition
$$
P\otimes S^2(V^\*) = P\otimes (P \oplus W) 
= \Lambda^2(P)\oplus S^2(P)\oplus P\otimes W.
$$
Clearly, $\mu\bigl(\Lambda^2(P)\bigr) = 0$ (these are the relations you are calling the 'Koszul relations'), so we need to examine the other two pieces. It is easy to check that $\mu$ is surjective, so, indeed, there is a $3$-dimensional kernel of the map
$$
\mu: S^2(P)\oplus P\otimes W\to S^4(V^\*).
$$
It's obvious that $\mu$ is injective on $S^2(P)$ and it's easy to check that it's injective on $P\otimes W$. The two $\mu$-images intersect in a space of dimension $3$ that is invariant under $S\_3$, and it is easy to see from this that the kernel $K$ splits as an $S\_3$-module as a sum $K\simeq \mathbb{C}\oplus W$. Hence, there is one relation that is invariant under the action of $S\_3$ and the complement is an irreducible $S\_3$-module of dimension $2$. You can write them out without difficulty, but they all involve a significant number of terms. 


