Is my husband who has degree from India eligible to claim American Opportunity Tax credit in US for a Post graduate course? This is his first degree in US.

No, because he has completed the first 4 years of post secondary education. 

Graduate work is eligible for the less generous (and non refundable) Lifetime Learning Credit. 

https://www.irs.gov/individuals/llc

No. Your husband cannot claim the American Opportunity credit if he has completed four or more years of post-secondary education.

To be eligible for AOTC, the student must:

• Be pursuing a degree or other recognized education credential
• Be enrolled at least half time for at least one academic period beginning in the tax year
• Not have finished the first four years of higher education at the beginning of the tax year
• Not have claimed the AOTC or the former Hope credit for more than four tax years
• Not have a felony drug conviction at the end of the tax year

The law does not specify that the four years have to be in the US, so he is ineligible for the AOTC if he has completed four or more years of post-secondary education anywhere in the world.