Extension if Indian visa issued for Czech republic

What you have must be a Czech visa. No. Indian embassy has nothing to do with a visa issued by Czech authorities.

No I am a Belgian national, it's my sister who wants an extension of visa

My sister has found a partner who is ready to have a live in relationship, on this basis can she get a Belgian visa

Your sister's partner can go to his local commune and ask what is the procedure to apply for a legal cohabitation.

Thanks, means she can stay back if her legal cohabitation is accepted

Only the commune can give you an official answer. This is not going to be very straightforward. In my knowledge, your sister needs a 'visa to declare a legal cohabitation in Belgium' obtained from a Belgian consulate. I am not sure if another schengen countries visa can be used to declare legal cohabitation in Belgium. This is what your sister's partner will have to ask the commune.

Moreover there are many strict requirements to be eligible to declare cohabitation.
Example: To apply for residence based on cohabitation, the partners must know each other for at least two years, have met in 3 different occasions with a total of 45 days, or have lived together for one year. There must be strong documentary evidence to prove this and every document presented as proof must be dated and verifiable.

There are plenty of resources and experience based testimonies available by google search that you need to go through, do your homework and research, and as I said, approach the commune for official answers.