Given a 4-cycle free graph $G$, can we determine if it has a 3-cycle in quadratic time?
Yes, this is known. It appears in one of the must-cite references on triangle finding...

Namely, Itai and Rodeh show in SICOMP 1978 how to find, in $O(n^2)$ time, a cycle in a graph that has at most one more edge than the minimum length cycle. (See the first three sentences of the abstract here: http://www.cs.technion.ac.il/~itai/publications/Algorithms/min-circuit.pdf) It is a simple procedure based on properties of breadth-first search.

So, if your graph is 4-cycle free and there is a triangle, their algorithm must output it, because it cannot output a 5-cycle or larger.