Are quotation marks appropriate when my POV character’s head and heart argue?

Sure, if the character is indeed debating himself out loud. Internal debates, however, should be treated like normal interior monologue, but with each “on the other hand” given it’s own paragraph. In some situations, one “side” of the argument can be put in italics (especially if there is mental illness or the spiritual realm involved.) Italics aren’t too popular, either, so try the whole thing in straight narrative first.

In case an example might help, here’s an excerpt from one of my unsold manuscripts, Son of Kristos, that shows how I handled a serious internal debate my hero had where his head and his heart are duking it out. The first thought is given direct and italicized because “your” doesn’t translate well. It’s speculative fiction and out of context, so some of the references won’t make sense to you, but the technique used should still be clear enough:

The pirate sneered at Meleon. “Now, am I goin’ to get out o’ here nice and slow with your partner, or are we goin’ to have us some fun killin’ hostages?”

Remember your training. The threat has to be real or they won’t back down.

But it was Diangelina.

He’d want to be the one in this position. It was for the greater good of society.

But it was Diangelina.

The more peacekeepers followed regulations properly, the fewer wolves bothered to take hostages, and the fewer innocent civilians who died. No one would take hostages if it wasn’t effective.

But it was Diangelina.

“Well, peacekeeper, what’s it goin’ to be?”

There had to be another way.