There are a variety of synonyms that can be used in place of "whiny conversation" depending on the exact tone and context. "Complaining dialogue," "petulant chatter," and "grumbling discussion" all convey a sense of dissatisfaction or annoyance being expressed verbally. "Mopey exchange," "grousing confab," and "sulky back-and-forth" all suggest a negative or sulky attitude being conveyed through conversation. "Moaning discourse," "woeful dialogue," and "bitter chat" all indicate a sense of sadness, disappointment, or frustration being communicated. Ultimately, the choice of word or phrase to replace "whiny conversation" will depend on the specific nuances and underlying emotions of the situation being described.