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.
A simple text search of the Bible on BibleGateway returns 3 references in the King James Version:
And it came to pass the third day, which was Pharaoh's birthday, that he made a feast unto all his servants: and he lifted up the head of the chief butler and of the chief baker among his servants.
But when Herod's birthday was kept, the daughter of Herodias danced before them, and pleased Herod.
And when a convenient day was come, that Herod on his birthday made a supper to his lords, high captains, and chief estates of Galilee;
-- Mark 6:21 (Referencing the same event as the Matthew passage above)
One example is omitted by the other answer (argument from silence inherent therein notwithstanding).
Job 1:4-5 And his sons would have a feast every man on his [birth]day at his own house, and would call for their three sisters to eat and drink with them. And when the days of the feast were concluded, Job would sanctify them, rising early in the morning and offering a burnt offering for them all, "in case," said he, "my sons have sinned, and have blessed God in their hearts." And so Job would do for each of their [birth]days.
(It's debated/debatable whether "blessed" is a euphemism for "cursed" here or not, or an expression of remorse or repentance—the former is more probable, and seems to be a convention used elsewhere in Job (1:11); it could have been an idiom of his day that has been lost to time.)