you can say it this way:
"Maligayang kaarawan Inay"
let's break it down into pieces:
the 'ng' after the adjective 'maligaya' is actually a particle (na), which I don't think has an english equivalent. thus, the right word is "maligaya na" contracted to "maligayang".
"Inay" (mom) - the formal term for mother though is "Ina" but usually, "Inay" is likely used.
additional 'birthday terms / phrase':
"ano'ng kahilingan mo?" (what is your wish.)
"ilang taon ka na?" (how old are you?)
"anong handa mo?" [what do you have (food) for your birthday?]
"may regalo ako sa iyo." (I have a present for you.)