According to Golumbic [1], Hajös proposed the following problem in 1957 (translation by Golumbic):
Given a finite number of intervals on a straight line, a graph associated with this set of intervals can be constructed in the following manner: each interval corresponds to a vertex of the graph, and two vertices are connected by an edge if and only if the corresponding intervals overlap at least partially. The question is whether a given graph is isomorphic to one of the graphs just characterized (Hajös [1957, p. 65, translated by M.C.G.]).
Golumbic also discusses how interval graphs were related to a question in biology made by Benzer [2] in 1959.
[1] Golumbic, Martin Charles. Algorithmic graph theory and perfect graphs. Vol. 57. Elsevier, 2004.
[2] Benzer, Seymour. "On the topology of the genetic fine structure." Proceedings of the National Academy of Sciences of the United States of America 45.11 (1959): 1607.