Both sitting on your lap, and seeing you get your beard or hair trimmed, are experiences that reassure your son that nothing will go wrong while he gets his hair cut.
Something that helped with our boys was finding a barber who specializes in younger children; these businesses may have smaller chairs, colorful bibs or tools, a television in the corner to distract, and/or a more cheerful demeanor. Any of these could make the overall experience more relaxing.
(Sadly, when my youngest was two, our nearby "kids haircut" business closed down. At that point, the older (three) was able to sit still long enough for a reasonable trim, but the younger sibling wouldn't deal with a "grownup" barber even when sitting on his father's lap. My last resort option was watching YouTube videos on basic haircuts, which kept my son looking respectable enough until he was older and could sit more calmly.)
If you need to resort to holding him, you can try turning your son sideways on your lap, so his arm is against your chest rather than his back, or even sit him facing you. This will give the barber better access to his hair, and you can also distract your son a little more when talking to him face-to-face (or face-to-side-of-face).