OK - unless they happened to put in a beam long enough for a standard opening but just made the opening narrow, the header beam is going to have to be replaced to reach the full width plus the support studs. You might luck out and find the beam goes the entire garage width - that changes it from a replace beam to maybe a strengthen beam job, which would cost a bit less. Because you have living space over the garage door, this job means supporting the entire section of house on a temporary stud wall during the swapout, and probably replacing the beam and moving (and maybe adding to) the supporting studs at both sides, a new garage door of course, and assorted trim and drywall and replacing some siding and painting and so forth.
The closest I have done to this is raising a 7 foot door to 8' for an 8' wide door, and that ran $3000 roughly, but only involved adding a panel to the door and added track sections for added height. You are doing basically the same thing just wider rather than higher, so you will be neding an entirely new gaarage door. Unless there is plumbing or structural issues not evident, I would say your budget should be $3500-4,000 range , and maybe you will be pleased if it comes in in the $2000's, which it might well easily do if the beam can be strengthened instead of replaced.
You need a general contractor for this to coordinate the trades, and be sure he and his subs are licensed, insured, AND bonded ! Search the List (in green banner bar above) for local contractors, their ratings and reviews. I would expect to have to talk to maybe as many as ten to get 2-3 to bid on it, because this is a tedious, somewhat high liability job for relatively little $, so many contractors will pass on it.