A well made wood staff should have the same grain structure as a bow. The grain running vertical end to end, runoffs or crazy grain in it would make a staff with weak points in it. If the staff was used heavily, took a lot of sharp impacts from other staffs there could be damage invisible to the eye.
If you can follow the grain end to end it might make a bow, I mean people have made bows with tool handles. If your staff is red oak, know that red oak is far from an ideal english longbow wood. Pretty much every single ELB made from red oak I was able to find has been backed with something stronger in tension like hickory or bamboo.
You won't know until you try, odds are good for ruining the staff, but if you no longer care for it or have a use for it and it's just eating you up inside to try to make a bow from it, figure out which way the grain runs in it, plane it flat parallel to the grain on what will become the back, and back it with bamboo or hickory and try to make a bow from it.
It would be cheaper to go out and find a straight grained board and probably achieve the same thing without ruining your staff though.
