Both the Force Missile and Quarter staff say "For your combat check..."
Neither of them add dice to a check, so they cannot add on to each other.
They both replace your combat check with what is printed on the card. You are changing "strength die" with "strength die +1d6". The base die is not changing, but the entire skill is.
It doesn't make much thematic sense either to be able to beat them over the head with a staff at the same time you are casting a spell.