Hi Linda, welcome,
Yes you absolutely can and should apply a finish to slate tiles. Search for slate dressings and finishes at your local tile store. Someone might help you but you failed to indicate where you are.
Tile 4 You LLC Troy, MI
KERDI Shower Specialist ... DITRA Installs ... Product-Method suitability consultation. I have NEVER made a mistake, I thought I did once...........but I was wrong! - PRODUCTS ADVERTISED WITHIN MY POST ARE NOT ENDORSED BY ME!