*This answer was deleted and then updated.*

Let
$$g(n,m)=\sum\limits_{i=0}^{n}\sum\limits_{j=0}^{m}\binom{i+j}{j}\binom{n-i+m-j}{m-j}\alpha^i\beta^j$$
I conjecture that
$$g(n,m)=\binom{n+m}{m}f(n,m)$$
Here is the PARI prog to verify this conjecture:

    f(n, m) = if(n==0 || m==0, (m==0)*(1 - a^(n+1))/(1-a)+(n==0)*(1 - b^(m+1))/(1-b)-(n==0 && m==0), a*n/(n+m)*f(n-1, m)+b*m/(n+m)*f(n, m-1)+1)
    g(n, m) = sum(i=0, n, sum(j=0, m, binomial(i+j,j)*binomial(n-i+m-j,m-j)*a^i*b^j))
    test(n, m) = g(n, m)==binomial(n+m, m)*f(n, m)
    n=6; x=sum(i=0, n, sum(j=0, n, test(i, j)))

But is there a way to prove it?