Are "ties" considering "successful" checks in Firefly Adventures?
Yes, you will need to reroll.

On page 4 of the rulebook:


  To make a Skill Test, roll the number of dice pictured and add all your available Skill Points of the appropriate type to your dice’s total. If your dice roll plus your Skill is equal to or higher than the Skill Test’s Target, it’s a Success. If it’s lower, it’s a Failure.


The rulebook clearly defines rolls as having a binary outcome, success or failure, and groups ties as a success for the player. In the case of a tie with the Martial Artist (i.e.: a success), you would need to reroll.