Given a number field K and a polynomial $$f(z) \in K[z]$$ , one can naturally construct a finite directed graph G(f, K) whose vertices are the K-rational preperiodic points of f, with an edge $$\alpha \rightarrow \beta $$ if and only if $$f(\alpha ) = \beta $$ . The dynamical uniform boundedness conjecture of Morton and Silverman suggests that, for fixed integers $$n \ge 1$$ and $$d \ge 2$$ , there are only finitely many isomorphism classes of directed graphs G(f, K) as one ranges over all number fields K of degree n and polynomials $$f(z) \in K[z]$$ of degree d. In the case $$(n,d) = (1,2)$$ , Poonen has given a complete classification of all directed graphs which may be realized as $$G(f,\mathbb {Q})$$ for some quadratic polynomial $$f(z) \in \mathbb {Q}[z]$$ , under the assumption that f does not admit rational points of large period. The purpose of the present article is to continue the work begun by the author, Faber, and Krumm on the case $$(n,d) = (2,2)$$ . By combining the results of the previous article with a number of new results, we arrive at a partial result toward a theorem like Poonen’s—with a similar assumption on points of large period—but over all quadratic extensions of $$\mathbb {Q}$$ .