We will give a example of countable infinite set of lines with every 2 lines intersect at a latice point. Construct, the lines $y=nx+n^2$ for all $n\in \mathbb N$.We need to show the following 3 facts: No 2 lines are parallel. Any 2 lines intersect at a latice point. No 3 lines concurr. First point is easy to proof, since slope of each lines are different. By solving equation, it is easy to show that $y=mx+m^2,y=nx+n^2$ intersects at $(-m-n,-mn)$. If 2 different lines meat at same point on $y=nx+n^2$ then we must have $-kn=-mn$ by comparing the $y$-co-ordinate of the intersection point.This is a contradiction.Hence no 3 lines concur. $\blacksquare$