Yes there are entrance exams for fashion designing in government colleges.
The popular entrance exams for fashion designing are :
1) NIFT- National institute of fashion technology accepts this entrance test
2)NID- national institute of design accept this entrance test.
3) CEED- IIT Bombay accept this entrance test
4)UCEEd- IIT Bombay accept this entrance test
5)AIFD- Army institute of fashion design, Bangalore accept this entrance test
6)FDDI- Footwear design and development institute
These all are entrance test of government colleges
You can apply on all the entrance test and then According to your result you will get admission.