Yes! You can do ME In Safety engineering course after a BE in Mechanical Engineering. So you have nothing to worry about just focus on the entrances such as GATE etc.

Top ME/MTech in Safety and Fire Engineering:

IIT Kharagpur - Indian Institute of Technology

OPJS University, Rajgarh

Vel Tech Rangarajan Dr Sagunthala R and D Institute of Science and Technology, Chennai

Sona College of Technology, Salem

Shri Rawatpura Sarkar University, Raipur

For more colleges visit the link below:

https://engineering.careers360.com/colleges/list-of-safety-and-fire-engineering-colleges-in-india

To know more about GATE 2022 visit the link below:

https://engineering.careers360.com/exams/gate

I hope this answer helps. All the very best for your future endeavors!