I've never received vested benefits so I'm not sure how that part works but I do use Fidelity for trading stocks and mutual funds so if you're sure you're ready to sell, I can lay out the basic procedure for you.
In your account, go to the positions tab, select the stock, and click "sell." You can choose to sell all shares at once as you suggested or sell a portion of them and hold on to the rest if you believe the company still has room to grow. Under the "order type" drop down you'll probably have some different options. A market order will execute at the next available price during trading hours. (Trades can only be executed while the market is open. The trade will execute once it reopens). Market orders can be somewhat risky if the stock is volatile since you can't be exactly sure what price you'll get for the shares.
Alternatively, you could execute a limit order. With this option, the order will not execute for less than the limit price you specify. This means you'll have to wait for the price of the stock to reach your limit but it also means you'll know in advance exactly how much you're getting for your shares. There are other order types (stop loss, stop limit, short sell, etc.) which I won't get into but I would encourage you to research.
Once you execute the trade, you'll have to wait for it to be filled at which point you'll receive a notification. You can also check the status of your trade under the "activity and orders" tab. It will say something like "order to sell 999 shares of Acme Corp. successfully executed at $10.59.
You're now ready to request your check, though it would probably be simpler to just do an ACH transfer to your bank account. Click on transfer up at the top and, assuming you want the check, click "See more transfer options" and on the next page, select "send me a check" and follow the instructions.
Congratulations! You've successfully converted stocks to cash. Keep in mind that all trade activity will be reported to the IRS. Please don't forget to file your taxes! If you're dealing with a significant amount of money, I would highly recommend seeking professional help. I am in no way, shape, or form a professional financial adviser and this is not to be taken as financial advice.