We can even assume the lattice is triangular: for each hexagon, add a point in its center and connect it to the six vertices of the hexagon. The conclusion holds for this larger lattice as well:
Assume WLOG that we can find points A,B belonging to this lattice s.t. if O is the origin of the plane, OB is obtained from OA through a rotation of angle π2 around O. This is equivalent to finding integers a,b,c,d s.t. a+bεc+dε=i, where ε is a solution to x2+x+1=0 (a+bε represents the point A, while c+dε - the point B). In turn, this implies that we can find rationals u,v with u+vε=i. This is claerly absurd, since eliminating u between u+vε=i,u+vˉε=−i gives an irrational value for v.