Cavil means to object when there's little reason to do so. It's a trivial objection.
In Thomas Hardy's Far from the Madding Crowd, Troy flirts and compliments his way into Miss Everdene's heart:
I am too gratified to get such a mark of your friendship at any price to cavil at the tone. I DO, Miss Everdene, care for it. You may think a man foolish to want a mere word — just a good morning. .