= Solution
For an integral <binary quadratic form> $[a,b,c]=ax^2+bxy+cy^2$ with $a>0$ and $D=b^2-4ac<0$, the <reduced positive definite binary quadratic form> convention is $|b|\le a\le c$, with $b\ge0$ on either boundary $|b|=a$ or $a=c$. Since $|D|=4ac-b^2\ge3a^2$, the <enumeration of reduced binary quadratic forms> for $D=-23$ has $a\le\sqrt{23/3}<3$. The integer $b$ is odd. For $a=1$, only $b=1$ is retained and $c=6$; for $a=2$, both $b=\pm1$ give $c=3$. Thus the complete list is
$$
\boxed{x^2+xy+6y^2,\qquad2x^2+xy+3y^2,\qquad2x^2-xy+3y^2.}
$$
A proper representation of a prime uses a primitive representing vector. Completing that vector to a matrix in $\operatorname{SL}_2(\mathbb Z)$ transforms a representing form to $[p,b,c]$, so its <discriminant of a binary quadratic form> gives $b^2\equiv-23\pmod p$. Conversely, choose an odd $b$ satisfying this congruence and put $c=(b^2+23)/(4p)$. This is integral, and $[p,b,c]$ is positive definite and represents $p$ at $(1,0)$. This proves the <discriminant criterion for prime representation by a binary quadratic form> here.
For $p\ne23$, <quadratic reciprocity> gives $(-23/p)=(p/23)$. Including the ramified prime, the answer is therefore
$$
\boxed{p=23\quad\text{or}\quad p\bmod23\in\{1,2,3,4,6,8,9,12,13,16,18\}.}
$$
For example $[23,23,6]$ supplies a proper representation of $23$. Reduction transfers every such representation to one of the three displayed forms.
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