How to concatenate IIR filter (Butterworth) with a Matlab function and evaluate the integral
The functions fdesign and design are for digital filters. Use butter instead to obtain the filter coefficients of an analog butterworth filter. Then use the function freqs to evaluate the (Laplacian) transfer function of that filter at given frequencies.

To use integral I would first create a function that calculates the transfer function of the concatenation of filters A and B. For example:

function H = filter_cascade(w, v)
    % transfer function of A
    H_A = ((1i.*w).^2)./(.1+1i.*w./v);

    % the other transfer function
    [b,a] = butter(3, 0.3);
    H_B = freqs(b, a, w);

    % The transfer function of the cascade of A and B
    H = H_A .* H_B;

    % Return the magnitude square of the transfer function
    H = H.*conj(H);
end


Then pass a handle to this function to integral like this:

v = 0.1;
q = integral(@(w)filter_cascade(w,v), 0, 1)./pi;


Note that I have changed the upper limit of the integral to one.