In Trajan, can a player get a bonus tile with zero votes on the senate track?
Yes

From the Rulebook, page 9:


  Ties are broken in favor of the player with the higher
  position on the senate track, in case of the same position
  in favor of that player whose disc is higher up in the
  stack. 


So, if all discs are still at the starting point, the order of discs on the stack matters for determining Consul and Vice-Consul.