First, substitute the solution x= -2, y= 11 into the first equation.  You will get then

    11 = m*(-2)^2 + n*(-2) + n,

or, equivalently,

    11 = 4m - n.    (1)



Next, substitute the solution x= -2, y= 11 into the second equation.  You will get then

    11 = n*(-2)^2 - m*(-2) + n,

or, equivalently,

    11 = 5n + 2m.    (2)



Equations (1) and (2) form the system of 2 equations in 2 unknowns m and n to solve. 
I will re-write the system in the standard form

    4m -  n = 11      (1')

    2m + 5n = 11      (2')


To solve the system, from equation (1') express  n = 4m-11  and substitute it into (2').  You will get

    2m + 5*(4m-11) = 11


Simplify and solve

    2m + 20m - 55 = 11

    22m            = 11 + 55 = 66

      m                      = 66/22 = 3.


Finally, from n = 4m-11 you find  n = 4*3-11 = 12-11 = 1.


ANSWER.  n = 1;  m = 3.