We present a targeted search for blazar flux-correlated high-energy ({epsilon}_{nu}_>~1TeV) neutrinos from six bright northern blazars, using the public database of northern hemisphere neutrinos detected during "IC40" 40-string operations of the IceCube neutrino observatory (2008 April to 2009 May). Our six targeted blazars are subjects of long-term monitoring campaigns by the VERITAS TeV {gamma}-ray observatory. We use the publicly available VERITAS light curves to identify periods of excess and flaring emission. These predefined intervals serve as our "active temporal windows" in a search for an excess of neutrinos, relative to Poisson fluctuations of the near-isotropic atmospheric neutrino background, which dominates at these energies. After defining the parameters of an optimized search, we confirm the expected Poisson behavior with Monte Carlo simulations prior to testing for excess neutrinos in the actual data. We make two searches: one for excess neutrinos associated with the bright flares of Mrk 421 that occurred during the IC40 run, and one for excess neutrinos associated with the brightest emission periods of five other blazars (Mrk 501, 1ES 0806+524, 1ES 1218+304, 3C 66A, and W Comae), all significantly fainter than the Mrk 421 flares. We find no significant excess of neutrinos from the preselected blazar directions during the selected temporal windows. We derive 90% confidence upper limits on the number of expected flux-associated neutrinos from each search. These limits are consistent with previous point-source searches and Fermi GeV flux-correlated searches. Our upper limits are sufficiently close to the physically interesting regime that we anticipate that future analyses using already-collected data will either constrain models or yield discovery of the first blazar-associated high-energy neutrinos.