Can I just rinse the knife & cutting board, instead of washing it, after cutting washed vegetables?

As a general food safety rule/best practice, you should still perform active washing, rather than passive rinsing, even for fruits and vegetables.

Fruits and vegetables are still vectors for food borne illness and cross contamination. Wide scale recalls of vegetables due to listeria and e. coli are not uncommon. Active washing with soap and water will combat this. A quick swipe with a soapy cloth followed by a rinse will clean up much better than rinsing alone.

Fruits and vegetables also contain things such as sugars, which passive rinsing may not fully clean. This can result in spoilage and/or mold growth on your cutting board. A great demonstration of this would be to cut beets, which have both relatively high sugar and an accompanying red color, and passively rinse up vs actively washing. Washing will clean up much better. The same is true for other fruits and vegetables, though the lack of red dye makes it less easy to notice.