The two sets I am hoping to intersect are the set of triangle numbers and quarter-squares. From those linked resources I have the generating function for both sets.
$$f_{A000217}(x) = \frac{x}{(1-x)^3} $$ $$f_{A002620}(x) = \frac{x^{2}}{((1-x)^{2}\cdot(1-x^{2}))}$$
I'm only starting to grasp generating functions and their uses, but is it possible to use them to find the generating function of their intersection?
Edit: After doing some more digging, I searched the OEIS for the opening few co-members (1, 6, 36, 210...) and found this sequence, which may be the solution but I have not validated.
$$f_{A002620}(x) = \frac{x}{((1-x)\cdot(1+x)\cdot(1-6\cdot x+x^2))}$$