Can I turn off my iPhone 13 and charge it wirelessly?

Sorry to tell you, but that is close to the worst possible strategy to save the battery. The only worse strategy would be to wait until the charge reaches zero and the phone turns off before charging it. Why? Because the phone is always using energy, even when “sleeping”. When you charge it some of the energy of running the phone comes from the external power source, rather than the battery, thus saving battery energy for when you need it. But if you charge it when the phone is off there is never a time when the energy to run the phone does not come from the battery. Then when you turn the phone back on it must catch up with all of the routine “housekeeping” that the phone did not do when it was powered off, so you actually haven’t saved any battery energy, all you have done is to put that ennergy usage off until you turn the phone back on.

The absolute best way to get maximum use on a charge, as well as slow the decline of battery capacity long term is to enable Optimized Battery Charging (Settings/Battery/Battery Health) and charge the device overnight, every night. The battery will fast charge to 80%, then pause. During the nighttime pause the phone will use mains power instead of battery power, allowing the battery to “rest”, and thus reducing the need to charge the battery quite as often. The phone will resume charging to reach 100% when you are ready to use your phone; it will “learn” your usage pattern. If you enable iCloud Backup (Settings/[your name]/iCloud - iCloud Backup) the phone will back up overnight also, assuring that you can never lose more than the current day’s updates. Here's more information→About Optimized Battery Charging on your iPhone - Apple Support

It has to be on to charge it wirelessly. Why do you want to turn it off to charge it?