Can you sell stock certificates at a bank?

No, you cannot sell stock certificates directly at a bank. In his digital and demat world, few banks have come up introducing their own demats. For example, ICICI bank has started ICICI Direct, SBI has started SBI Demat etc.,
By approaching them you can find a way to sell your stocks. They can convert your physical shares or stock certificates into digital mode and record them in Demat and then process for selling your stocks.