I read in the verse that Passover is in the, “…month of Aviv.” But I cannot seem to find it in the Jewish calendar. When is the Jewish month of Aviv?
The Hebrew word aviv (Heb. אביב) actually translates as the season of “spring.”
We find mention of the “month of Aviv” several times in the Biblical writings. The example you brought from Exodus 13:4 reads:
Today you are going out, in the month of Aviv.
In his classic commentary Rabbi Shlomo Yitzchaki, also known as Rashi, explains why it uses spring in this context:
Now do we not know in what month they went out? Rather, this is what Moses said to them, “See the loving-kindness that He bestowed upon you, that He took you out in a month in which it is suitable to go out, when there is neither heat nor cold nor rain.”
So aviv is not the name of a month, but rather one of the four seasons.
We still need to understand why the verses keep on referring to Passover as being in the “month of Aviv,” the month of spring.
The sages explain on the verse (Deuteronomy 16:1), “Guard the month of Aviv, and make [then] the Passover offering,” that the use of “spring” is telling us that we should ensure that Passover will always coincide with the right season, during the aviv, spring.
For a more detailed explanation, see How Does the Spring Equinox Relate to the Timing of Passover?