I assume that the "conversation" you refer to is the talk that the frog has with the nightingale during their first and subsequent meetings, and what these conversations reveal about the characters and their intent.
When the frog first approaches the nightingale, he says:
" ... You see,
I'm the frog who owns this tree
In this bog I've long been known
For my splendid baritone
And, of course, I wield my pen
For Bog Trumpet now and then"
He asserts his possession of the tree and stresses his authority, thus making the nightingale feel like an invader, an outsider who has illegitimately laid claim to his dominion. Furthermore, since he's "long been known", he is the one everyone in the bog is familiar with and "of course," he is a skilful master of his craft with his "splendid baritone". The fact that he "wields" his pen "now and then" further emphasises his authority.
These statements obviously put the nightingale at a distinct disadvantage. The frog makes her feel uncomfortable for invading his space and makes her immediately feel inferior. The frog's assertions illustrate his pomposity and supercilious nature. He is arrogant and boastful. Added to this, he is also sly, because he realises that the nightingale is much more talented than he and therefore, not only must he put her in her place, but that he could also exploit her talent to his benefit.
The next few lines illustrate that the nightingale lacks self-confidence. she is flustered as shown by her stammer. The frog recognises this and pounces. He criticizes her singing, saying that it's "not too bad", when he knows that it is excellent. Her singing lacks a "certain force". The frog manipulates her into believing that she's not good enough.
These lines not only illustrate the nightingale's lack of self-esteem, but also that she is quite naïve and gullible. She deems the frog's appraisal as a compliment, when it is far from that. When she states that the song is at least her own, the frog heartlessly dismisses even that as, "not much to boast about". In doing this, he does not allow the nightingale to reclaim at least some dignity. He insults her to substantiate his later claim that, "with me you'll be a winner."
The foolish nightingale takes the frog's word for it and unquestioningly accepts his invitation to instruct her for payment. It is ironic that she deems him a "Mozart" and believes that his offer is "a fairy tale", when it is everything but. She will soon realize what a nightmare it actually is. The lines reveal how extraordinarily manipulative and ruthless the frog is and how unbelievably naïve the nightingale.
The frog says:
""Well I charge a modest fee."
"Oh! " "But it won't hurt, you'll see"
These words are tragically ironic since his 'fee' turns out to be far from 'modest'. And, as far as the 'hurt' is concerned, the nightingale eventually pays with her life.
If the creatures in "The Frog and the Nightingale" had a conversation after the poem, provided, of course, that the nightingale had not died, I would have relished the opportunity to see the nightingale address the frog's pretentious behavior. For this conversation to occur, the nightingale would have to have some sort of epiphany that the frog's behavior was inexcusable, but from the way Seth has drawn the characters in the poem, I do not believe that the frog would ever admit to any wrong doing on his part. The nightingale could talk to him until she burst a vein in her head (like in the poem), and the frog would still stubbornly hold to the belief that he was the better singer and acted altruistically on the bird's behalf.