I'm not sure if it is a nuance of the word 「ごねる」 which means to make difficulties or to grumble.
Here is the sentence containing the word.
ごね出した【だした】 means "started to grumble." 出す is one of the syntactic compound verb elements. After the masu-stem of a verb, it means "to begin to ～", similarly to ～始める. See: What is the difference between 出す and 始める when used as a suffix?