“In the seventh month, on the first day of the month, you shall observe a sacred occasion: you shall not work at your occupations. You shall observe it as a day when the horn is sounded.” (Num. 29:1)
Jewish tradition adopted the new moon of the seventh month as the start of the new year, Rosh Hashanah. The oddity of having the New Year begin in the seventh month comes from the fact that two calendrical systems are reflected in the Bible: one in which the new year begins in the fall, and one beginning in the spring, as the Babylonian year did.
The calendar in this section of Numbers is an interesting hybrid: the year begins in the fall, but the month is reckoned according to the spring beginning. (Note that the months here are numbered, not named. This is typical of the Babylonian reckoning.) One reason for highlighting the seventh month rather than the first might be an analogy with the Sabbath. Just as the Sabbath offering amounts to twice the daily offering, so the offering for the “seventh” new Moon is twice the usual offering for the new moon.
In any case, despite the Babylonian reckoning beginning the year in the spring, an autumnal inception for the year is very old. It is reflected in Exodus 23:16, an early text, and in an even earlier inscription known as the Gezer calendar (10th century BC, found at Tel Jazari 18 miles NW of Jerusalem).