I would imagine it depends on insurance, with the answer being no in most cases.
Edit: my limited experience with private insurance is apparent. It would seem, as noted below, there are indeed non-medicare insurances that do cover services provided by supervised fellows working toward licensure. I imagine you'd just have to check with each individual insurance co.
Medicare will not pay for postdoctoral level psychotherapists. Most private insurances, to my understanding, will. I know this because pretty much all the available psychotherapists at my last institution (CA) were postdocs, and it was impossible to find therapists for Medicare patients because they could only be seen by a faculty level person and that just wasn't very available. But there were tons of insured patients seeing postdocs and apparently billing their insurances for the work. I did not directly supervise psychology trainees myself so not sure how the billing worked exactly.
Certain services provided by your employee (or a fellow employee) may also be eligible for Medicare payment, but check your state law for exceptions and requirements.
www.apaservices.org
" Interns and Postdoctoral Graduate Students
Medicare does not allow for services provided by interns or postdoctoral graduate students to be billed under the “incident to” provisions. Medicare considers services furnished by interns or postdoctoral graduate students to be training and therefore not eligible for Part B funding.