In this work we study quantum algorithms for Hopcroft’s problem which is a fundamental problem in computational geometry. Given (n) points and (n) lines in the plane, the task is to determine whether there is a point-line incidence. The classical complexity of this problem is well-studied, with the best known algorithm running in (O(n^{4/3})) time, with matching lower bounds in some restricted settings. Our results are two different quantum algorithms with time complexity (\widetilde O(n^{5/6})). The first algorithm is based on partition trees and the quantum backtracking algorithm. The second algorithm uses a quantum walk together with a history-independent dynamic data structure for storing line arrangement which supports efficient point location queries. In the setting where the number of points and lines differ, the quantum walk-based algorithm is asymptotically faster. The quantum speedups for the aforementioned data structures may be useful for other geometric problems.