You can, but it needs to penetrate deeper into into the wood or you'll tend to steel wool it off during finish application.
I recommend a good water or alcohol based stain.
Surprisingly, the old Birchwood-Casey Walnut Water Stain works very well, but you have to thin it down with alcohol or it stains too dark, too fast.
Brownell's sell several other water based stains.
One of the best looking stains for walnut is one of the "Pre-1964" stains. This gives the wood a slight reddish color like the old American rifles had, and that Winchester used prior to 1964.
Minwax used to sell the Antique Oil Finish with a walnut stain mixed in, but seem to have discontinued it.