Were birthdays celebrated by believers anywhere in Holy Scriptures? Is there any mention of a birthday celebration among Old Testament Israelites or New Testament Christians in the Holy Scriptures?
Were birthdays celebrated by believers?  The answer would turn on one's definition of "celebrate".

Does "celebrate" mean only the inclusion of cake, ice cream, balloons, streamers, presents, friends, and family singing songs, showing pictures, opening presents, pinning the tail on the donkey, and blowing out candles?  Or how "celebrate" means only about dancing girls and beheading of enemies?  Or would it be more mundane, as in I "celebrated" my birthday last month, realizing my age on the earth?

If the latter, than of course, they "celebrated" their birthdays.  For example, they knew Jesus was 12 years old when He visited the temple.  They knew John the Baptist was 30 years old when he became a priest.  One can't know these things unless someone "celebrates" one's birthday.

Again, we don't know exactly what "celebrate" meant because it is not recorded exactly how they "celebrate" those birthdays, but we do know they didn't pretend the dates didn't exist or in fact were not material to their lives.  Birthdays were obviously "celebrated".

Subsequent to the New Testament, there is the account of the Martyrdom of Polycarp.  He knew he was 86 when he was tried in the fire and martyred.  Of course once you are dead, no one is observing your birthday on earth, but what is interesting is their thinking they might observe his birthday into death.

Other answers posted quote scripture examples.  There were obvious birthday "celebrations" too deviant, just as there might be today.  But there are other birthday "celebrations" in the right spirit.

PS To add, according to this site, Isaac the first Jewish baby was weaned at 24 months.  Abraham and Sarah had a great feast.  Tradition places it on his second birthday.