We study the computational complexity of finding the geodetic number of a graph on chordal graphs and interval graphs. A set S of vertices of a graph G is a geodetic set if every vertex of G lies in a shortest path between some pair of vertices of S. The Minimum Geodetic Set (MGS) problem is to find a geodetic set with minimum cardinality of a given graph. We show that Minimum Geodetic Set is fixed parameter tractable for chordal graphs when parameterized by its tree-width (which equals its clique number). This implies a polynomial-time algorithm for k-trees, for fixed k. Then, we show that Minimum Geodetic Set is NP-hard on interval graphs, thereby answering a question of Ekim et al. (LATIN, 2012), who showed that Minimum Geodetic Set is polynomial-time solvable on proper interval graphs. As interval graphs are very constrained, to prove the latter result, we design a rather sophisticated reduction technique to work around their inherent linear structure.