We present the results of a search for pair production of a heavy toplike (t′) quark decaying to Wq final states using data corresponding to an integrated luminosity of 5.6  fb−1 collected by the CDF II detector in pp collisions at s√=1.96  TeV. We perform parallel searches for t′→Wb and t′→Wq (where q is a generic down-type quark) in events containing a lepton and four or more jets. By performing a fit to the two-dimensional distribution of total transverse energy versus reconstructed t′ quark mass, we set upper limits on the t′t′ production cross section and exclude a standard model fourth-generation t′ quark decaying to Wb (Wq) with mass below 358 (340)  GeV/c2 at 95% C.L.