The terms and conditions say "First Eligible Stay can be at any participating property during the Promotion Period to trigger Bonus Point eligibility for any future stays."
Considering booking a stay nearby so I can earn the bonus points for the stay next month. Does the first eligible stay have to be at a Hyatt Place or can it be another Hyatt brand?
Your first stay does not have to be at a Hyatt Place. It can be at any Hyatt brand. My first stay for this promo was at a Hyatt House, for example.