Question marks are used at the end of direct questions:
Will she ever take me seriously?
The sentence you’re using isn’t a direct question. In fact, it’s a statement detailing the narrator’s thought. As a result, it should end with a period. These kinds of questions are called indirect questions. There’s certainly a question in there, but it’s not stated directly. Here’s another indirect question:
The interviewer asked how the applicant would handle an emergency situation.
These kinds of statements come up often in exchanges that are summarized. If you wanted to make it a direct question, ending with a question mark, it should read something like this:
"How would you handle an emergency situation?" the interviewer asked.
Don’t confuse indirect questions with compound sentences that begin with statements but end with questions. For example:
Shelly wasn’t late, but would he still be at the diner when she arrived?
This is different than a compound sentence that ends with an indirect question:
Shelly wasn’t late, but she wondered if he’d still be there when she arrived.