No. Only parents, spouses, children over 21 and siblings. Your cousin can petition for a greencard for his parents who can then petition for their sibling who should be your parent who can then petition for you. However this is likely to take a serious amount of time, as sibling petitions take over 10 years. Marriage to a U.S. Citizen, as long as it is genuine, is the quickest route. If you entered the U.S. illegally, in most cases you can not adjust your status in the U.S.
A cousin relationship is not qualified a relationship for a family based petition. The following family relationships qualify for family based petitions:
1. Spouse/Minor Child/Parent of a US Citizen
2. Adult Child (Married or Unmarried of a US Citizen)
3. Spouse/Unmarried Minor and Adult Child of Lawful Permanent Resident
4. Brothers/Sisters of Adult US Citizens
Having said that, your cousin can be useful to you in at least two ways.
First, if your cousin is the closest US relative, then your next best option is a nonimmigrant visa that will keep you lawfully. A cousin with financial resources can make a great financial sponsor for i.e. a student visa or visitor visa application. As a graduate of US university, you will be eligible for one year of employment authorization opening the door to a work based temporary visa as your next logical step.
Second, looking more long term, your cousin can help you network and find a US employer that would be willing to sponsor you. Employment based immigrant visas ("green card") is the best long term option for someone who does not have close US family relationships.
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