Early Quenya
nosta
noun. birth, birthday
nosta-
verb. to give birth to; to cause
malto ísier i nosta - qarda
*by those from whom this birth was known - less
puita-
verb. to beget
There are more words but they are from Tolkien's earlier conceptional periods. Tolkien likely changed these words as he evolved the aesthetics and completeness of the languages. You may even find languages that Tolkien later rejected. Do not mix words from different time periods unless you are familiar with the phonetic developments.
Below: Tolkien's Late Period (1950–1973). These are from Tolkien's late conceptual period — his most settled forms of the languages. Prefer these over the earlier ones above when composing new text.
A noun appearing as ᴱQ. nosta “birth, birthday” in the Qenya Lexicon of the 1910s derived from the early root ᴱ√NŌ “become, be born” (QL/66).
Neo-Quenya: As the root √NŌ survived in Tolkien’s later writings with very similar meanings, I’d retain ᴺQ. nosta for purposes of Neo-Quenya, but I’d limit its use to “birth”. For “birthday” I’d use ᴺQ. nostarë, a neologism that has been in use for so long that I cannot identify where it came from.