I'll do a little letter re-assigning and assume that you're after prime elements

in quadratic integer rings

corresponding to quadratic fields

, where we restrict

and thus there is a natural ordering of the primes.

I think it will be easiest to characterise primes when the rings happen to be unique factorisation domains (UFD), namely D = -1, -2, -3, -7, or -11. So when D = -1, it is congruent to 3 (mod 4) and we have the Gaussian integers. Because it is a UFD, we have that primes and irreducibles are equivalent, so no need to worry about that. So the first prime integer in the Guassian integers is 3. The next ones are 7, 11, 19, and 23. You can see some info here

Gaussian Primes
and here

id:A002145 - OEIS Search Results
A nice result is that these are precisely the primes over the integers that are not the sum of two squares. More info here

Math Forum - Ask Dr. Math
For choosing D other than -1, -2, -3, -7, or -11, we have to be careful about the distinction between prime and irreducible. Every prime is irreducible, but the converse is not necessarily true. Pay special attention to the example given here

Irreducible element - Wikipedia, the free encyclopedia
The example I mean is: "In the quadratic integer ring

, the number 3 is irreducible but is not a prime since 9 can be written as

and 3(3)."

Another well-studied ring is the Eisenstein integers obtained by letting D = -3. More info here

Eisenstein integer - Wikipedia, the free encyclopedia Eisenstein prime - Wikipedia, the free encyclopedia