Goethe and Hegel both get some character-driven exposition in this book, but Hegel is about twenty years younger than Goethe, so the interactions focus more on the contemporaneous groupings of Goethe-Schiller-Fichte and later Schelling-Hegel than direct interaction between the two. Goethe as a poet is generally viewed more through the lens of literary criticism as opposed to the philosopher Hegel who is generally studied with the analytical rigor of philosophy, considering his position as one of the premier post-Kantian German idealists.
The two share much ideological agreement, like when both offered great praise for Schiller's Letters on the Aesthetic of Education of Man, but while Hegel generally quotes Goethe with veneration, Goethe seems generally suspicious of Hegel's philosophy -- in one meeting offering open criticism to Hegel about his dialectic, for example.
One work that goes into detail on both thinkers would be the late Walter Kaufmann's first volume of his trilogy Discovering the Mind, in which he offers an opinionated criticism of Kant and Hegel's obscurity and obfuscation by contrasting it with Goethe's thematic clarity.