Sometimes such similar sounding words do share a close similarity, being common variations of the same theme. That is not the case here, except in the case of picking at and pecking at. Although they are metaphors for two different physical actions, the sense they convey is the same.

The verb "pick" has multiple meanings. Most often it means "choose" or "collect", but one other meaning is similar to the primary meaning of "peck." The verb "peck" usually describes a bird striking something with its beak, or it can also describe a person suing a sharp tool to strike something. They both also share one other less frequently used meaning. If you eat something really slowly in very small bites, it can be described as "picking at" or "pecking at" your food.