how to design LQR controller with time-variable parameters ?

Yes, a MATLAB Function block would indeed do the trick. Suppose you make a function:

function output = myLQR(error,param)

Where param is the the value of that one element of A (let's say, A(5,5)) passed in as a time-varying signal to the MATLAB function. So, in the function, you can say something like:

A(5,5) = param;

Then, the command to design an LQR contoller given A and B matrices, and Q/R/N noise covariance matrices, is as follows (or look at the lqr documentation for other ways to call it).

K = lqr(A,B,Q,R,N)

Finally, you want to multiply the resulting K matrix with your error signal, and that's your control law!

output = K*error;