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.