Yes. Corporations too pay LTCG and STCG. It is the nature of income which decided LTCG AND STCG, not the class of company.
e.g Profit/loss of sale of mutual fund , profit/loss on sale of investment property, profit/loss on sale of fixed assets ( provided there are no assets in the block)
These are examples of some incomes which could be LTCG /STCG assuming these are not the principal business activities of the corporation.