A Jewish Palestinian circa 30AD would know their age, and would celebrate some days in relation to their age, for example a Bar Mitsvah. But whether or not they would celebrate their birthday I don't know.

As to whether the knowledge of the date of ones birth would be feasible, or if it would be culturally relevant to know ones age, there are many, many examples of Jewish people knowing their age, going back to Abraham listing his and his wife's ages, and on to kings, all the patriarchs, and in the specific context of your question, Jesus circa 30 AD is said to be about 30 years old, likely as there is a requirement in Numbers 4:3 for a priest to be 30 years old or older. Given that there were age requirements in the culture, given that there were numerous feasts and celebrations tied to the calendar (i.e. all the feasts were tied to the calendar, and were spread around the year), and that the torah from the very beginning considers ages to be important enough to record age at death and age at birth of the first child, it is likely that a Jewish person would be able to know the day of their birth, and keep track of their age, but whether or not there was any kind of a celebration of it I have no idea and hope someone else does.