Liz,
That would probably depend on the clinic -
I did a preliminary search for you and did find a couple who states on their websites that they do accept Medicaid. I will list them below for you, but just as a precaution, I would give them a call just to make sure.
Somerset Treatment Services
118 West End Avenue
Somerville, NJ 8876
908.722.1232 ext 2
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New Horizon Treatment Services Inc
132 Perry Street
Trenton, NJ 8618
609.394.9888
http://www.nhts.net/
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New Brunswick Counseling Center
320 Suydam Street
New Brunswick, NJ 8901
732.246.4025
http://www.newbrunswickcounselingcenter.org/
I'm sure there are others to check out, but this is just a few to get you started.