Shruti,
You need to specify Neumann conditions with the NeumannValue[] function or by specifying the derivative directly like this. You also must specify some initial conditions for x and q (Dirichlet conditions).
s = NDSolve[{D[x[n, t], t] == D[x[n, t], {n, 2}] - q[n, t] x[n, t],
D[q[n, t], t] == -q[n, t] Exp[x[n, t]], x[n, 0] == 0.01,
q[n, 0] == 0.01, Derivative[1, 0][x][0, t] == 0.0,
Derivative[1, 0][x][90, t] == 0}, {x[n, t], q[n, t]}, {n, 0,
90}, {t, 0, 10}]
Plot3D[x[n, t] /. s, {n, 0, 90}, {t, 0, 10}, PlotRange -> All]
I made up some conditions because you did not specify any:
x[n, 0] == 0.01, q[n, 0] == 0.01
Now the problem solves. You can specify more interesting initial conditions for x and q as long as you respect the derivative constraints at the two ends.
Here is an example in which I made up a complex function for x over the range of n=0 to n=90. The Sin function is flat at n=0 and n=90 (the two peaks) and dips in between. I also added a slope of 0.005 between n=0 and n=90 to make it more interesting. Note that I also needed to add the slope 0f 0.005 to the derivatives to make everything consistent.
s = NDSolve[{D[x[n, t], t] == D[x[n, t], {n, 2}] - q[n, t] x[n, t],
D[q[n, t], t] == -q[n, t] Exp[x[n, t]],
x[n, 0] == Sin[(n + 90/4)* 2*Pi/90] + 0.005*n, q[n, 0] == 0.01,
Derivative[1, 0][x][0, t] == 0.005,
Derivative[1, 0][x][90, t] == 0.005}, {x[n, t], q[n, t]}, {n, 0,
90}, {t, 0, 10}]
Plot3D[x[n, t] /. s, {n, 0, 90}, {t, 0, 10}, PlotRange -> All]
If you look at the plot, you can see its all consistent.
Regards
Neil
