We will need to find all the prime ideals of $\mathbb Z[\sqrt{-5}]$ and then find each $P$-primary ideal for each prime ideal $P$.
Let $p$ be a prime in $\mathbb Z$. The non-zero prime ideals of $\mathbb Z[\sqrt{-5}]\cong \mathbb {Z}[X]/(X^2+5)$ are $$\{(p,g(\sqrt{-5})):g(X)\text{ is an irreducible factor of }X^2+5\text{ in }\mathbb F_p[X]\}.$$ See this post for a justification of why this is true.
Finding the primary ideals is a little more tricky. Read through this MathOverflow question and the comments.
If $X^2+5$ is irreducible in $\mathbb F_p[X]$ then $(p,g(\sqrt{-5}))$ is the same as $(p)$, and in this case all the $(p)$-primary ideals are given by a power of $(p)$.
When $X^2+5$ is not irreducible mod $p$ but $p^2$ does not divide $\text{disc}(X^2+5)=-20$ then again all the $(p,g(\sqrt{-5}))$-primary ideals are powers of $(p,g(\sqrt{-5}))$.
This leaves only one more case. Since $X^2+5$ is reducible modulo $2$ with irreducible factor $X+1$ and $2^2 \mid \text{disc}(X^2+5)$ then all the powers of $(2,\sqrt{-5}+1)$ are still $(2,\sqrt{-5}+1)$-primary ideals, but there may be more $(2,\sqrt{-5}+1)$-primary ideals that are not of this form.
Unfortunately I am not sure how you would go about finding all the remaining $(2,\sqrt{-5}+1)$-primary ideals.