Connecticut gives you the same combat zone tax breaks the federal government gives. Base pay, hostile fire and imminent danger pay, bonuses, school loan repayments and money from selling accrued leave are excluded from tax while you serve in a combat zone or a contingency operation, with exceptions for commissioned officers. You also get a 180 day extension to file and pay while you are deployed, or while you are hospitalized for injuries or illness received in a combat zone or contingency operation.
There is nothing to apply for. Leave the excluded pay off your Connecticut return, and if you need the extension, file when the 180 days end rather than by the normal deadline. The IRS Tax Exclusion for Combat Service Fact Sheet explains which pay qualifies.
Your orders and Leave and Earnings Statements showing the combat zone or contingency operation service dates, and hospital records if you are using the extension because of an injury or illness.