GATE

Yes you can get. The M.Tech admissions in COEP Pune are based on GATE and Non Gate criterias. The first preference is given to a candidate having the valid GATE score. The departments may specify additional requirements over and above these minimum requirements. All the Non-GATE candidates will have to undergo an entrance test conducted by department in which he/she is applying. Passing in this test will be mandatory for admission.
Eligibility criteria : Bachelors degree in Engineering/Technology or equivalent in an appropriate area, with a
minimum of First Class/60% marks or CGPA of 6.5 on a scale of 10 or equivalent (CGPA of 6.00 or equivalent in case of SC / ST).

Yes, you can do M.Tech in Biomedical Engineering. The eligibility criteria of IIT Mumbai is given below:
1. Eligibility criteria (Qualifying Disciplines) : B.E. /B.Tech. /AMIE or equivalent in Biomedical Engineering (BM), Biotechnology (BT), Chemical Engineering (CH), Computer Science and Engineering/ Information Technology CS Electrical Engineering (EE), Electronics/Telecommunications Engineering (EC), Engineering Physics (EP), Instrumentation Engineering (IN), Mechanical Engineering (ME), Metallurgy & Materials Science (MT), Pharmaceutical Technology (PY), Other Engineering (ZE).
2. You have to give Bio-medical Engineering GATE paper with code no. BM for admission in M.Tech courses.