Yes you have the option, but those sentences have different meanings.
The first sentence, written in the simple past, is about what he did during the call. The time in the past that encompasses the call is the time in the past that encompasses what he said. Both are in the past to the reader but contemporaneous to each other.
When he called her, he did not tell her about quitting his job.(Implication: When the call ended, he still had not told her.)
The second sentence, written in the past-perfect, is about what had already happened when the phone call was made. The time in the past that encompasses the call is preceded by the everything before that, which includes what he said or did not say about the job. Both are in the past to the reader but what he said (nothing) about quitting is in the past to the call.
At the time that he called her, he had not yet told her about quitting his job. (Implication: If he told her during the call, it was the first she heard of it from him. But we don't know what happened during the call from this sentence.)
1. My alternates are to display what would be understood from each sentence but are not better constructions.
2. We do not know from either sentence if he has already quit his job or if he is contemplating or has decided to quit.