I moved in 2021 for a new job. In 2022 I filed my receipts and was paid for the moving expenses. Do I need to count the moving expenses in 2021?

if you are a w-2 employee, it doesn't matter - the expenses are not tax deductible in any event.  If your employer reimbursed you for those moving expenses, I suspect that will appear as 'wages' in your W-2 as those reimbursements are taxable income to you. 

If they are on your 2022 W-2, you will claim that income when you file your 2022 return in early 2023.

Under the TCJA, moving expenses are not deductible, and they are not eligible for tax-free reimbursement from your employer.  If you were reimbursed in 2022, you should expect to see the moving expenses included in your taxable wages on box 1 of your W-2 (as if they paid you a taxable bonus that you happened to use for moving).