dont instead of que

No, dont would be incorrect here. Dont replaces "de + noun" and introduces a relative clause when the verb or expression in the clause requires an indirect object introduced by de. In the sentence you post, the relative pronoun que stands for the direct object.

Here is a sentence as an example for dont:

Tu connais mon ami Alain , dont je t'ai parlé ? -- You know my friend Alain about whom I told you?

Notice that dont in this sentence replaces "parler de qqn".