Yes, it does happen but there was no specific time in which Naruto started having feelings for Hinata. It all sort of happened over time due to a series of, what you could call, events.

When Toneri kidnapped Hanabi Naruto, Hinata, Sakura, Sai, and Shikamaru were deployed on the mission to rescue her. Later, while the team got caught in a Gatekeepers genjutsu that dredges up old memories, the red scarf (that Hinata made and gave to Naruto, but was later ripped)that was in the process of being repaired got loose from her bag and wrapped around Naruto, sharing her memories and feelings with him.

Left in shock after witnessing how long Hinata had loved and acknowledged him, Naruto became fiercely protective of her, even seemingly bashful in her presence.