You may say '(O)tanjoubi omedetou.'
お誕生日おめでとうございます => 'Otanjoubi omedetou gozaimasu' is polite and 誕生日おめでとう => 'Tanjoubi omedetou' is very casual/informal way to say 'happy birthday' in Japanese.
Tanjo-bi omedeto!
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Otanjōbiomedetōgozaimasu is the Japanese pronunciation of the English phrase, "Happy Birthday". It can be quite hard to pronounce so break it down into syllables.