Can we find the largest intersecting subfamily of convex polygons in quadratic time?
As Discrete lizard pointed out in a comment, I misinterpreted the statement of &quot;sharing a common point&quot;, which I read as &quot;pairwise sharing a common point&quot;.
You can thus ignore the below answer.  I will delete it later.

I'm not an expert in this area but I do actually believe that the problem you are talking about is surprisingly enough NP-complete.
The problem, known as Maximum Clique in convex planar sets (graph class: Conv)  was proved NP-hard by Kratochvíl and Kuběna [1] who write the following:

We show that complements of planar graphs have intersection representations by convex sets in the plane, i.e., for every planar graph, one can assign convex sets in the plane to its vertices in such a way that two of the sets are disjoint if and only if the correspondning vertices are adjacent. This fact has a complexity consequence — it follows that the problem of determining the clique number of an intersection graph of convex sets in the plane is NP-hard. We note that the complexity of this problem for intersection graphs of straight line segments in the plane is unknown.

[1] Kratochvíl, Jan, and Aleš Kuběna. &quot;On intersection representations of co-planar graphs.&quot; Discrete Mathematics 178.1-3 (1998): 251-255