I don't believe so.
There is an order of priests that serve in the temple (a rotation of 24) but the starting priest varies from year to year.
Now these are the divisions of the sons of Aaron. The sons of Aaron;
Nadab, and Abihu, Eleazar, and Ithamar. But Nadab and Abihu died
before their father, and had no children: therefore Eleazar and
Ithamar executed the priest's office. And David distributed them, both
Zadok of the sons of Eleazar, and Ahimelech of the sons of Ithamar,
according to their offices in their service. And there were more chief
men found of the sons of Eleazar than of the sons of Ithamar; and thus
were they divided. Among the sons of Eleazar there were sixteen chief
men of the house of their fathers, and eight among the sons of Ithamar
according to the house of their fathers.
Thus were they divided by
lot, one sort with another; for the governors of the sanctuary, and
governors of the house of God, were of the sons of Eleazar, and of the
sons of Ithamar. And Shemaiah the son of Nethaneel the scribe, one of
the Levites, wrote them before the king, and the princes, and Zadok
the priest, and Ahimelech the son of Abiathar, and before the chief of
the fathers of the priests and Levites: one principal household being
taken for Eleazar, and one taken for Ithamar.
Now the first lot came
forth to Jehoiarib, the second to Jedaiah, The third to Harim, the
fourth to Seorim, The fifth to Malchijah, the sixth to Mijamin, The
seventh to Hakkoz, the eighth to Abijah, The ninth to Jeshua, the
tenth to Shecaniah, The eleventh to Eliashib, the twelfth to Jakim,
The thirteenth to Huppah, the fourteenth to Jeshebeab, The fifteenth
to Bilgah, the sixteenth to Immer, The seventeenth to Hezir, the
eighteenth to Aphses, The nineteenth to Pethahiah, the twentieth to
Jehezekel, The one and twentieth to Jachin, the two and twentieth to
Gamul, The three and twentieth to Delaiah, the four and twentieth to
These were the orderings of them in their service to come
into the house of the LORD, according to their manner, under Aaron
their father, as the LORD God of Israel had commanded him. (1
Chronicles 24:1-19 KJV)
Luke 1 identifies that Elisabeth conceived sometime after Zacharias' experience with the messenger's visit, but we're not told exactly how long after.
The references to Elisabeth being hid five months in Luke 1:24 suggests the two references to "the sixth month" are specifically speaking of a point in her pregnancy term.
Luke 1 never identifies that Mary conceived prior to the birth of John the Baptist, so we have no way to gauge the age of Jesus in relation to John.
In Luke 2, the reference to the days of purification being over is referring to the number of days a mother is considered clean after childbirth according to Leviticus 12. It has no bearing on the time of year.
And the LORD spake unto Moses, saying, Speak unto the children of
Israel, saying, If a woman have conceived seed, and born a man child:
then she shall be unclean seven days; according to the days of the
separation for her infirmity shall she be unclean. And in the eighth
day the flesh of his foreskin shall be circumcised. And she shall then
continue in the blood of her purifying three and thirty days; she
shall touch no hallowed thing, nor come into the sanctuary, until the
days of her purifying be fulfilled.
But if she bear a maid child, then
she shall be unclean two weeks, as in her separation: and she shall
continue in the blood of her purifying threescore and six days. And
when the days of her purifying are fulfilled, for a son, or for a
daughter, she shall bring a lamb of the first year for a burnt
offering, and a young pigeon, or a turtledove, for a sin offering,
unto the door of the tabernacle of the congregation, unto the priest:
Who shall offer it before the LORD, and make an atonement for her; and
she shall be cleansed from the issue of her blood. This is the law for
her that hath born a male or a female. (Leviticus 12:1-7 KJV)