We has : Case 1$ m^2-mn-n^2+1=0$ $ \Delta=n^2+4(n^2-1)=5n^2-4$ So $ 5n^2-4=k^2$ Then root of this equation is $ n_0=1,n_1=2,n_{r+2}=3r_{n+1}-r_{n}$ $ k_0=1,k_1=4,k_{r+2}=3k_{r+1}-r_n$ $ m_k=\frac{n+k}{2}$ $ n\geq 1981$ so $ n=610,k=1364,m=987$ Case 2 :$ m^2-mn-n^2-1=0$ $ \Delta=5n^2+4$ so $ 5n^2+4=k^2$ Root of equation is 1.$ n_0=1,k_0=3$ $ k_{r+1}=9k_r+20n_r$ $ n_{r+1}=4k_r+9n_r$ $ \Max{n}=377,k=843,m=610$ 2. $ r_0=3,k_0=7$ $ k_{r+1}=9k_r+20n_r$ $ n_{r+1}=4k_r+9n_r$ $ max{n}=987,k=2207,m=1597$ So $ \max\{m^2+n^2\}=1587^2+987^2$

For reference, another solution can be found here.