No, you don’t report stock purchases or stock you hold on your tax return.

Also, if the stock is held at a broker, as is typical, you don’t report it as a payer of dividends, either; the dividends are paid to you by the broker.

So, the IRS generally doesn’t know about any specific stock you own until you sell it. (They do know if you own dividend-paying stocks, because they get a report of dividends paid, but they don’t get a report of which stocks you own.)