Yes. The spread at exercise of an NQSO is considered wages for the purposes of Social Security and Medicare taxes. Together these are sometimes called FICA taxes or payroll taxes. Both the employee and the employer pay these taxes on the income spread recognized at the time of exercise.
The wage base for Social Security tax is $160,200 in 2023. Social Security tax does not apply to yearly income in excess of the annual income ceiling. The rate of Social Security tax is 6.2%, giving a maximum possible Social Security withholding of $9,932.40 in 2023.
The Medicare tax rate, which has no wage ceiling, is 1.45% for single filers with yearly taxable income of $200,000 or less and for married joint filers with yearly taxable income of $250,000 or less. The rate is 2.35% for single taxpayers whose yearly taxable income is more than $200,000 and for married joint filers whose yearly taxable income is more than $250,000. (For details, see a related FAQ.)
Railroad Employees
Under a ruling by the US Supreme Court in Wisconsin Central Ltd. v. United States (2018), the spread at exercise of an NQSO is not taxable compensation under the Railroad Retirement Tax Act (RRTA) because stock options (also probably RSUs and other forms of stock compensation) are not considered "money remuneration" as the term is meant in the statute (written in 1937). The RRTA provides funding for pension-like benefits to railroad employees instead of Social Security retirement benefits. The spread at exercise would still be subject to the standard income tax rules. For more on the decision, see a commentary in The myStockOptions Blog.
Alert: For the application of President Trump's Aug. 2020 executive order creating a payroll-tax holiday for the employee portion of Social Security, see the explanation of it in this website's Ask The Experts section.