It depends. Moving company quotes may be negotiable within reason based on the scope of the job. While many costs may be fixed in nature (such as labor costs for movers) you may be able to discuss details of specific services based upon the total estimated package for your move. Be sure to reach out to a member of the moving team in advance if you have any questions or concerns associated with any costs; you may be able to use your knowledge of the move to impact the final estimate (such as details pertaining to the weight of items and the types of stairs at facilities).
Most moving companies expect that you are comparing cost estimates and would rather work with you to find mutually agreeable terms than to lose your business altogether, so it can’t hurt to open up the discussion.