If your lumber is pressure treated, that contains arsenic and other chemicals that are poisonous. Any other type of lumber would be ok to plant edibles near.

If the fence is older than three years, it is probably pressure treated with chromium copper arsenate (CCA). This chemical was voluntarily removed from the industry at that time, but other dangerous chemicals have replaced it.

Nothing sensational about it! If you were to read the warnings on CCA lumber, you'd freak out.