I learned something new today! I deal with a lot of Bermuda grass in my yard and, in my opinion, it should be classified as invasive. Around some of my fruit trees, I’d like to kill off the grass and plant wildflowers. This publication recommends a treatment of herbicide while protecting the fruit tree trunks with a wrapping, like aluminum foil
I wouldn’t spray your trunk with herbicide if you can help it, but ultimately it won’t hurt it, depending on the formulation of herbicide.
You have different formulations of herbicides. Some formulations, namely ester formulations, can be sprayed on a tree trunk to kill a tree but have to be sprayed with a carrier which is usually diesel fuel or other oil. Other formulations, like salt formulations, are generally safe to spray on tree trunks and nothing will happen. Most home use herbicides, think Glyphosate, are gonna be in formulations that are safe to spray on a tree trunk.