With the widespread use of GPS-enabled mobile devices, an unprecedented amount of trajectory data is becoming available from various sources such as Bikely, GPS-wayPoints, and Uber. The rise of innovative transportation services and recent break-throughs in autonomous vehicles will lead to the continued growth of trajectory data and related applications. Supporting these services in emerging platforms will require more efficient query processing in trajectory databases. In this paper, we propose two new coverage queries for trajectory databases: (i) k Maximizing Reverse Range Search on Trajectories (kMaxRRST); and (ii) a Maximum k Coverage Range Search on Trajectories (MaxkCovRST). We propose a novel index structure, the Trajectory Quadtree (TQ-tree) that utilizes a quadtree to hierarchically organize trajectories into different quadtree nodes, and then applies a z-ordering to further organize the trajectories by spatial locality inside each node. This structure is highly effective in pruning the trajectory search space, which is of independent interest. By exploiting the TQ-tree data structure, we develop a divide-and-conquer approach to compute the trajectory "service value", and a best-first strategy to explore the trajectories using the appropriate upper bound on the service value to efficiently process a kMaxRRST query. Moreover, to solve the MaxkCovRST, which is a non-submodular NP-hard problem, we propose a greedy approximation which also exploits the TQ-tree. We evaluate our algorithms through an extensive experimental study on several real datasets, and demonstrate that our TQ-tree based algorithms outperform common baselines by two to three orders of magnitude.