According to R' Yehoshua’s tradition of the opinion of the Rabbis, we do not say that an animal gives birth after incomplete months, and therefore if it gave birth within its first year it must have become pregnant within 30 days of the discharge, which means the discharge was not indicative of a fetus and does not exempt the next offspring from being counted a firstborn. According to R' Yehoshua’s own reasoning, we say an animal gives birth after incomplete months, and consequently it is possible the animal became pregnant later than 30 days after the discharge and its term of pregnancy was shorter than normal. If so, the discharge does exempt the next offspring from firstborn status.
ואיבעית אימא: לא אמרינן יולדת, והכא במקצת היום ככולו קמיפלגי. לסבריה — אמרינן מקצת היום ככולו, לגמריה — לא אמרינן מקצת היום ככולו.
And if you wish, say that everyone accepts Ze’eiri’s statement and we do not say that an animal gives birth prematurely, and here, they disagree with regard to whether or not the halakhic status of part of the day is like an entire day: According to R' Yehoshua’s reasoning we say part of the day is like an entire day, and therefore it is possible for the goat to have become pregnant on the 30th day after experiencing the discharge and to give birth precisely 5 months later, just before the year ends. According to his tradition of the opinion of the Rabbis, we do not say part of the day is like an entire day, which means that the earliest possible birth is on the 1st day of the 2nd year.