Possibly. Council tax is usually paid in instalments across the year, so moving out partway through can leave you in credit — but whether it does depends on how far through the year you got and what the council works out you owed.
Ask your old council whether your account is in credit, how any credit will be refunded or put towards another bill, and when. Some councils send a form out with the closing bill that you need to complete, sign and return before the money is released, and it goes to whatever forwarding address they have on file. If the date they gave you passes and nothing's arrived, chase them.
Moved a while ago and never left a forwarding address? The credit may still be sitting there. Contact the council and ask.