NONLINEAR EQUATIONS WITH FSOLVE

I am not certain what you want to do.

This uses ‘T’ as a single vector entirely (rather than for each element):

P     =  35;
FCH40 = 100;
T     = [923 : 10 : 1123]';
X0    = [1; 10];

fcn = @(X) [(((P^2)*(X(1)-X(2)).*(3*X(1)+X(2)).^3)./((FCH40-X(1))*(3*FCH40-X(1)-X(2))*(4.01*FCH40+2*X(1))^2))-10.^((-11650./T(:,1))+13.076); 
            (((0.01*FCH40+X(2))*(3*X(1)+X(2)))./((X(1)-X(2))*(3*FCH40-X(1)-X(2))))-10.^((1910./T(:,1))-1.784)];

XS = fsolve(fcn, X0);

This considers each element of ‘T’ in a loop:

P     =  35;
FCH40 = 100;
T     = [923 : 10 : 1123]';
X0    = [1; 10];

fcni = @(X,Ti) [(((P^2)*(X(1)-X(2)).*(3*X(1)+X(2)).^3)./((FCH40-X(1))*(3*FCH40-X(1)-X(2))*(4.01*FCH40+2*X(1))^2))-10.^((-11650./Ti)+13.076); 
            (((0.01*FCH40+X(2))*(3*X(1)+X(2)))./((X(1)-X(2))*(3*FCH40-X(1)-X(2))))-10.^((1910./Ti)-1.784)];

for k1 = 1 : numel(T)
    XS(:,k1) = fsolve(@(X) fcni(X,T(k1)), X0);
end

figure(1)
plot(T, XS)
grid

The plot makes it easier to see the solutions.

Experiment to get the result you want.