QuickBooks records a reimbursed cost one of two ways, and the choice is made by a single setting: the reimbursement either posts as income alongside the expense, or posts back against the expense account as though the cost had never been incurred.
In QuickBooks Online the control sits under the gear icon, in Account and settings, on the Expenses tab, in the Bills and expenses section — the option reads "Track billable expenses and items as income." Turning it on offers a further choice: track everything in a single account, which QuickBooks names Billable Expense Income, or track across multiple accounts, which adds a checkbox to each income account so you can mark the ones eligible to receive billable expense income. Left off, a reimbursement posts back to the expense account it came from, and that expense reads as though the agency never incurred the cost at all.