In the United States, we always put periods and commas inside quotation marks, but twice in the last week I've seen writers break this rule when confronted with a quotation that ended with a question mark and also needed a comma after it to make the whole sentence work. Here's an example from this week's Grammar Girl podcast and article about Christmas carols:
The Christmas carol we're going to tackle today is "What Child Is This?", written in 1865 by William Chatterton Dix.
Omit the Comma Before an Attribution
Although it's not exactly the same situation, the AP Stylebook says that when you'd put a comma at the end of a quotation before the attribution, but the quotation ends in a question mark, you should omit the comma. ("Merry Christmas," Squiggly said. "How many days until Christmas?" Aardvark asked.)
Keep the Comma After a Title
The Chicago Manual of Style editors make a similar recommendation for attributions, but take a new stance on titles in their newest edition (16th edition, section 6.119): they recommend keeping the comma when a title ends with a question mark or exclamation point, as in the example above from the Grammar Girl podcast. Therefore, according to Chicago, which addresses the question most directly, the best way to write the sentence is as follows:
The Christmas carol we're going to tackle today is "What Child Is This?," written in 1865 by William Chatterton Dix.