We present a search for a new particle T' decaying to top quark via T' -> t + X, where X is an invisible particle. In a data sample with 4.8 fb(-1) of integrated luminosity collected by the CDF II detector at Fermilab in p (p) over bar collisions with root s = 1.96 TeV, we search for pair production of T' in the lepton + jets channel, p (p) over bar -> t (t) over bar + X + X -> lvbqq'b + X + X. We interpret our results primarily in terms of a model where T' are exotic fourth generation quarks and X are dark matter particles. Current direct and indirect bounds on such exotic quarks restrict their masses to be between 300 and 600 GeV/c(2), the dark matter particle mass being anywhere below m(T'). The data are consistent with standard model expectations, and we set 95% confidence level limits on the generic production of T'(T) over bar -> t (t) over bar + X + X. For the dark matter model we exclude T' at 95% confidence level up to m(T)' = 360 GeV/c(2) for m(X) <= 100 GeV/c(2).