We made a military move for the year 2022. When we settled in, we submitted a claim for the travel expenses. Is this countable for the government reimbursement part?

No, if you are in the military, don't include any payment you received from the government for the settlement of your travel claim. That is not considered reimbursement for moving expense purposes. Your only entry would be for an amount shown in box 12 of your W-2, identified with a code P (this will only be entered when you enter your W-2).

Please see Military Moving Expenses for additional information.