If you are asking a question, then yes. If you are simply telling people what you're wondering about, then it isn't a question and it should not have a question mark.
Some examples to help clarify:
I often wonder about the future.
I wonder what is for breakfast.
I wonder... who is that?
People will often say phrases like these as asking a question but its usage can be considered awkward for written English.
I am curious about your name?
That is pie?
And you are?
The question marks here are more flagging the intonation we use in everyday speech. You can also use an ellipsis:
And you are...
Your name is...
On tests you will see a blank line:
The answer is _____________
But for "I wonder..." I would stick to a period/full stop. When you say it out loud, feel free to say it much like you ask a question.