A billable expense is an expense that you have incurred on behalf of your client/customer for performing work, services, supplies. Billable expenses are reimbursable from your customer simply by billing them the incurred expenses when you send them an invoice.
Yes, you're correct. You can create a new bill and make this billable to your customer. I've attached a screenshot for your visual guide.
In case it's time to pay the bill, please read this article for more information: Enter And Pay Bills.
Please let us know if you need further assistance. I'd be glad to help.