Is there any pseudorandom function that outputs $n$ prime number of ring $Z_{N}$ where $N$ is Paillier public key
TL;DR: Any (pseudo-) random value from the desired range has a high chance of being invertible.

To understand this, first remember the criterion for a value $x$ to be multiplicatively invertible in $\mathbb Z_n$: $$\gcd(n,x)=1$$

Now remember how many such values exist, exactly: $$\varphi(n)=(p-1)(q-1)=pq-p-q+1$$

Now assume these non-invertible values are distributed somewhat uniformly over the set of all integers smaller than $n$. Then the chance that a value is invertible is exactly
$$\Pr[x\in \mathbb Z_n:\exists x^{-1}]=\frac{\varphi(n)}{n}=\frac{pq-p-q+1}{pq}=1-\frac{1}{p}-\frac{1}{q}+\frac{1}{pq}$$

Now remember that $p,q$ are large (otherwise the encryption could be broken), then $1/p,1/q$ and $1/(pq)$ are negligible and thus the probability of an integer being invertible is nearly 1.

As for constructing such values: Just generate bit strings of the appropriate size and discard them until they fit your limitations (e.g. $\leq n$), usually you will have to discard none or one integers this way.