It seems that this has been answered here by S. Mustonen: [PointsInGrid.pdf][1]

There $f(n,m)$ is denoted by $L_n(m)$ and a formula would be 
$$L_n(m)=\frac{1}{2}[f(m,n+1)-2f(m,n)+f(m,n-1)]$$
for $$f(m,k)=\sum_{\substack{-m<kx<m\\-m<ky<m\\(x,y)=1}}(m-|kx|)(m-|ky|)$$



  [1]: http://www.survo.fi/papers/PointsInGrid.pdf