This is ultimately a question of prescriptivism vs descriptivism. In common usage, "going to be" and "will be" are used in effectively identical cases, and "is" can often be heard in many of those same cases as well. But the English grammatical prescription does make a distinction between the three:
- "Is" should be used for the present tense. A birthday happening a month into the future would not qualify.
- "Will" should be used for either facts about the future ("the Sun will rise tomorrow"), promises about the future ("I will help you move house next week"), predictions about the future based on beliefs or opinions, or actions or events in the future that are decided at the current moment.
- "Going to" should be used for actions or events in the future that have either been decided beforehand, are about to happen immediately, or are predicted to occur (with less certainty than a fact) based on existing evidence.
As I said, the differences are extremely minor and quite often not followed in real-world speech, but if the test is looking for formality and pedantry, that's the distinction they're probably using. By those rules, "Your birthday's next month" is probably the most egregious violation, and the one they consider the most incorrect.
It's worth noting that by these same distinctions, the most correct answer would be "your birthday is going to be next month", since it's not a guaranteed fact (in a very sad way) but it's predicted based on existing evidence. However, that's also the option my own experience tells me is the least common in real usage.