No the brain cannot feel any pain because the brain contains no nociceptors which is responsible for feeling pain.
The body contains the nociceptors and those nociceptors send the signals to the brain which causes us to feel pain so the brain itself is unable to feel any pain.
But the skin that covers the brain on our head can feel pain because it contains those receptors.
So brain surgery can be done on the brain itself and you won't feel any pain when they cut into the brain although you're under anesthesia when having brain surgery.
The brain doesn't feel the pain but instead our bodies feel the pain when we get cut, hurt etc.