Hi, I am in the military I claim CT as residence but live in NM do I need to claim and pay tax for purchases in NM such as Internet being Active Duty stationed in NM?

You do not have to include the sales tax for things you purchased in New Mexico on your Connecticut state tax return for the use tax.  The use tax applies to items purchased and "USED" in the state for which no sales tax was paid.  Since you were in New Mexico, it does not apply.

Use tax is the tax you are required to pay when you use a taxable good or service in Connecticut on which you did not pay sales tax to a retailer. The use tax is complementary to the sales tax. Together, the sales and use taxes treat purchases of taxable goods and services that are used in Connecticut equally, whether the goods and services are purchased within or outside Connecticut.