There are still a lot of misconceptions about walnuts being vegetables when in reality they are not. Nuts are simply a hard-shelled form of fruit. You will find that there are many different types of nuts, including hickory nuts, chestnuts, hazelnuts, Brazil nuts, and walnuts.
A lot of “nuts” are actually just kernels or seeds, despite what many people may think. Nuts that are nothing more than seeds include cashews as well as Brazil nuts and pistachios. Walnuts in particular are nothing more than a hard-shelled fruit, putting to rest the notion that they are actually a vegetable.
The whole vegetable v fruit idea, most famously about tomatoes, is actually meaningless. Words often have multiple definitions depending on context. Fruit has a botanical and a culinary meaning. Certain things are fruits scientifically regardless of how we think of them. They may or may not be fruits in the world of cooking/nutrition. In the latter context tomatoes are not fruit, and neither are walnuts. But in contrast vegetable has no scientific meaning, only a culinary/dietary one. It is therefore meaningless to say “A is not a vegetable”, meaning it isn’t one scientifically. Nothing is a vegetable to science. Only in the context of talking about food does that word mean anything, and science has nothing to say about it.