No daycare teacher wants potty-trained children to pee themselves. I really wouldn't accuse her of lying.
When I have that age-group I insist that every child sit on the toilet before going outside. It's very, very common for kids to tell you they don't have to go but when they sit on the toilet, voila! If you don't "help" the kids remember to go by requiring them to visit the toilet, then yes, of course, someone is going to have to go literally two minutes after you take them outside or someone is going to pee themselves. Daycare is a much more fast-paced, stimulating environment than being at home and even VERY potty-trained kids can have accidents.
My suggestion is that you ask your child care provider to send your kid to the toilet before going outside. If he pees, great. If he doesn't, fine.
If you really think your child's day care provider is lying to you, you need to pull your kid out TODAY and find a new daycare. You do not leave your child with someone you don't trust.