Given operators X and Y acting on a separable complex Hilbert space H, an interpolating operator is a bounded operator A such that AX=Y. In this article, we investigate Hilbert-Schmidt interpolation problems for operators in a tridiagonal algebra and we get the following: Let ${pounds}$ be a subspace lattice acting on a separable complex Hilbert space H and let X=$(x_{ij})$ and Y=$(y_{ij})$ be operators acting on H. Then the following are equivalent: (1) There exists a Hilbert-Schmidt operator $A=(a_{ij})$ in Alg${pounds}$ such that AX=Y. (2) There is a bounded sequence ${{alpha}_n}$ in $mathbb{C}$ such that ${sum}_{n=1}^{infty}|{alpha}_n|^2<{infty}$ and $$y1_i={alpha}_1x_{1i}+{alpha}_2x_{2i}$$ $$y2k_i={alpha}_{4k-1}x_2k_i$$ $$y{2k+1}_i={alpha}_{4k}x_{2k}_i+{alpha}_{4k+1}x_{2k+1}_i+{alpha}_{4k+2}x_{2k+2}_i;for;all;i,;k;mathbb{N}$$.