We know that χ(G)>ω(G) is alwasys true for any graphs. So to prove the equality you just have to prove that χ(G)<=(G) is true as well. So you should admit that your graph is ω-colourable, it is enough to show an appropriate colouring using ω colours.

It is useful to think over what a clique means in that G complement graph and which vertices are in the same colour class.

clique: vertices that represents disjoint intervals

vertices in the same colour class: they represents intervals that have a common point (have a common intersection)

I hope that with the help of these hints you can solve the problem.