Turkish birthdays

I'm a NICU nurse in the US, and recently I've been taking care of little preemie Turkish twins. They're about to turn one month old, and I wanted to make a sign for their isolettes or a little card or something similar. Googling "happy birthday in Turkish" yields Doğum günün kutlu olsun! - is this correct? Can it be used for a one-month birthday, or does it denote being a year older? Is there something that more nearly means "Happy one month!"?

Thanks!