Conversation is interactive communication between two or more people. The development of conversational skills and etiquette is an important part of socialization. The development of conversational skills in a new language is a frequent focus of language teaching and learning. Conversation analysis is a branch of sociology which studies the structure and organization of human interaction, with a more specific focus on conversational interaction.
Both could be used.
However, the use of “whether” always implies at least one alternative.
So, if “whether” is used, the sentence should be either:
“They discussed whether they should build a hospital or not.”
or
“They discussed whether they should build a hospital or a nursing home.”