Refers specifically to the fruit of the walnut tree.
Mordí un ajonuez fresco que me ofreció un amigo en el mercado.
I bit into a fresh walnut that a friend offered me at the market.
El ajonuez que compré estaba tan fresco que no pude resistir comerlo de inmediato.
The walnut I bought was so fresh that I couldn't resist eating it immediately.