There are a couple of methods I've used to address these, however both will leave evidence, of which you may be able to mask.
Because the material is hardwood, an air nailer's nail will often leave a dimple, I'll pre-drill the board, the thickness of the board, about 1/2", near each edge, a size smaller than an 8P finish nail, at about a 30 degree angle, opposing each other, insert the nail, setting the head of the nail just enough below the board surface to accept putty stick.
This will leave greater evidence to address. There are glue kits specifically designed for
this repair which amount the a syringe and glue and you again drill holes in the board, insert the syringe and dispense the glue. Some of these require a weight to be placed and lift for a measure time, to insure adhesion.