You cannot export the Cost Code information along side the Bill of Quantities to Excel, you can however print it. Follow the below steps:
Ensure your project has been assigned cost codes
Reports, Estimating, Print Bill
Select the Item Buildup Option (Middle right)
Select the Setup button, select Resource Cost Code from the left and Add to the Right, OK
Select OK
This will give a print out of the Item details including the Buildup breakdown with the Cost Code.
Tip: There are other Excel exports which can include Cost Codes, check the Options of the report you want to see if Cost Code can be included.