I would like to know if it is possible to apply for a green card if I married in the other state from where I live in and if she is my first cousin.

Generally, if a foreign national entered the United States lawfully and with inspection, and becomes married to a U.S. citizen, the couple may succeed with a marriage-related adjustment of status case by which the foreign national spouse can become a U.S. Lawful Permanent Resident (get a "Green Card"). Generally, the federal government's USCIS will recognize a marriage as lawful if it was lawful where it took place, and so the validity of a marriage to a first cousin will be determined by the laws of the state where the marriage takes place. Of course, there are many other details that determine eligibility for this type of application process. It would be wise for you and the woman you plan to marry to consult with an immigration lawyer who, after learning all of the relevant information about you and the woman, would be able to advise about immigration eligibilities, options and strategies and would be able to offer legal representation in the often quite complex application process.

You might have issues with that. You also need to check the law where you got married: if such marriage is allowed under the state laws.