Short answer
The Azores have many traditional festivals, often tied to religious celebrations and local culture. Major ones include the Holy Spirit festivals (Festas do Espírito Santo) in spring and summer, and the Sanjoaninas in Terceira in June. Each island also has its own patron saint festivities. Check local calendars for exact dates, as they vary yearly.
General information, not legal or tax advice. Rules and amounts change: confirm with official sources.