I hope this isn’t too much of a silly question, but I’m curious about how your months work. I know that when it’s winter here, it’s summer for you, but are your calendar months the same as everywhere else? For example, when we’re in December and it’s freezing, is that when your summer starts? Do you still refer to it as June, or is it something different? I apologize if this seems like a ridiculous question—I just want to be able to explain it to my son, and I realized I’m not entirely sure myself. Thank you for your understanding!

No question is stupid, and it’s great that you’re asking for clarification! In Australia, the calendar months are the same as in most parts of the world. So, yes, they still call it December when it’s summer there.
To break it down a bit:
So when you’re experiencing winter in December, Australia is enjoying summer! Your son will probably find it interesting to learn about how different countries experience the seasons at different times of the year. If you have any more questions, feel free to ask!