What is the width of the mounting bracket and the weight of the TV?
Not knowing this, here are a couple of thoughts.
If you are handy, you can cut the drywall (i assume it is drywall) and screw plywood between the studs. This would give you a good base to screw into, then re-cover the plywood with drywall, plaster and paint and screw the mouning bracket into it. You need to consult your local building codes for fire ratings of materials needed.
If you cannot see behind the tv, you could also surface mount plywood to give you the same result (not very pretty though).
Depending on the weight of the unit you could put screws with large heads (i would add oversized washers as well) into the center stud to carry the weight of the unit and use drywall anchors to fill out the other fixture points.
Please consult your local hardware store on advice on the best hardware to use. I would bring the mounting bracket with you if you can.