In this paper, we study the following nonlocal nonautonomous Hamiltonian system on whole R (−∆) 1 2 u + u = Q(x)g(v) in R, (−∆) 1 2 v + v = P (x)f (u) in R, where (−∆) 1 2 is the square root Laplacian operator. We assume that the nonlinearities f, g have critical growth at +∞ in the sense of Trudinger-Moser inequality and the nonnegative weights P (x) and Q(x) vanish at +∞. Using suitable variational method combined with the generalized linking theorem, we obtain the existence of at least one positive solution for the above system. Mathematics Subject Classification (2010). Primary 35J50, 35R11, 35A15. Keywords. elliptic systems involving square root of the Laplacian, critical growth nonlinearities of Trudinger-Moser type, linking theorem.