The first searches for B^{0}→K^{+}π^{-}τ^{+}τ^{-} and B_{s}^{0}→K^{+}K^{-}τ^{+}τ^{-} decays at the LHCb experiment are conducted with pp collision data corresponding to an integrated luminosity of 5.4 fb^{-1}. The tau leptons are reconstructed using the τ^{+}→μ^{+}ν[over ¯]_{τ}ν_{μ} decay and the results are presented in bins of K^{+}π^{-} or K^{+}K^{-} mass. No signal is observed and upper limits are set on the branching fractions. The searches result in the first upper limits for B^{0}→K^{+}π^{-}τ^{+}τ^{-} decays outside the K^{*}(892)^{0} region in K^{+}π^{-} mass and the first limits for B_{s}^{0}→K^{+}K^{-}τ^{+}τ^{-} decays. The searches are recast into limits on the decays B^{0}→K^{*}(892)^{0}τ^{+}τ^{-} and B_{s}^{0}→ϕ(1020)τ^{+}τ^{-}, yielding 2.8×10^{-4} (2.5×10^{-4}) and 4.7×10^{-4} (4.1×10^{-4}) at the 95% (90%) confidence level, respectively. For the decay B^{0}→K^{*}(892)^{0}τ^{+}τ^{-}, this result improves on the current best upper limit by an order of magnitude.