Does options exercise increase a stock's trading volume?

No. Yes. It does not have to - it all depends whether the stocks you get are bought by the seller or whether he transfers you stock from his portfolio (i.e. I am long, sell you a call, then give you my shares). In the last case there is no transaction happening that hits the trading volume.

If satisfying the assignment/exercise of the contract results in transfer of shares to the counterparty, there is no stock exchange transaction since the trade occurs at the strike price between the two counterparties.

If one of the parties needs to acquire the shares in order to deliver or if one of the parties chooses to close the assigned shares then a transaction will occur on the stock exchange.