No.
If you are eligible for a HSA, i.e. you have a High deductible Health policy (HDHP), then your employer's contributions to your HSA are not taxable and are listed in box 12 of your form W-2 with code W.
You do not add them back to your income.
That's just it I've been questioning this ever since I started. When I look at my payslips I see the Employer contribution being added to my gross pay. Then it lists my "pre-tax" deductions my "employee taxes" then it lists "post tax deductions" and when I look at the YTD post tax deductions I see the employer contributions in the post tax deductions as well. Like I said this is a new company and when I reach out they are saying everything is correct.
Please look at your W-2 in box 12 for the code of W. This number should be the sum of what your employer contributed to your HSA PLUS what you contributed by means of payroll deduction. The code W amount is subtracted from Wages in boxes 1, 3, and 5 before the W-2 is printed.
Compare what you see on the paystubs and see if it matches the code W amount.
It's theoretically possible for the employer to try to make after tax contributions to your HSA, but these contributions would apply to your limit and if they aren't in your code W amount, then you would have to make a special entry in TurboTax.
So, ask again to make sure that the code W amount matches your paystubs, and then make sure that the code W was removed from Wages (you'll have to ask HR, because you can't tell from looking at the W-2).