Baptiste was stretchered off before half-time during Boro’s friendly with York following lengthy treatment on the pitch after being injured in a challenge by a triallist.
The 29-year-old defender moved to the Riverside Stadium on Monday from fellow Championship side Bolton on a three-year deal.
“I don’t like to speak about football after that,” Boro manager Aitor Karanka told the club’s website. “I lost the game after what happened to Alex.
“I don’t know exactly what has happened, but it looks like a serious injury, and he could be out for a long time. I think it is a complete break but we need to wait and see.
“It’s the worst way to start pre-season.”
Bradley Fewster scored 11 minutes from time to secure a 1-0 victory over the League Two side at Bootham Crescent.
“The attitude was good,” added Karanka. “York are a good team and it was another good session.”