This is more of a long commentary on the question.
This is a very interesting question, thank you OP.
This is a revised version of my answer.
Perhaps the question can be reworded this way.
Let $N$ be a positive integer. If $X$ be a set of points in the plane
that $|X|=N$ and no three of which lie on the same line, then denote
by $c_N(X)$ the number of fours of $X$ whose convex hull is a triangle.
What is the maximal value of $c_N(X)$?
This problem can also be reformulated as follows.
For a given integer $N$, find the maximum possible integer $\bar{c}_N$
such that any set of $N$ points in the plane in general position has
at least $\bar{c}_N$ convex quadrilaterals.
It is clear that
$$c_N+\bar{c}_N=\binom{N}{4}.$$
Further, we can observe that the number of convex quadrilaterals on the set $X$ of points in the plane in general position is equal to the number of crossings of the edges of the complete graph on the set $X$ if its edges are rectilinear.
Therefore the convex quadrilateral problem is equivalent to the following problem.
Find the rectilinear crossing number of the complete graph $K_N$,
i.e., to determine the minimum number of crossings in a drawing of $K_N$ in the plane
with straight edges and the nodes in general position.
We denote this number by the symbol $\bar{v}(K_N)$.
We see that $\bar{c}_N=\bar{v}(K_N)$.
It turns out to be a pretty old problem.
Here are some first values $\bar{c}_N=1,3,9,19,\ldots$ for $N=5,6,7,8\ldots$ (OEIS A014540). Quite a few values are known for small $N$ see here
To illustrate what we have said, here is one picture.
On the left is the configuration $X$ of $5$ points, for which $\bar{v}(K_5)=\bar{c}_5(X)=1$.
The configuration $X$ of $6$ points is shown on the right, for which $\bar{v}(K_6)=\bar{c}_6(X)=3$.
