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Active galactic nuclei (AGN) are promising candidate sources of high-energy astrophysical neutrinos, since they provide environments rich in matter and photon targets where cosmic-ray interactions may lead to the production of gamma rays and neutrinos. We searched for high-energy neutrino emission from AGN using the Swift-BAT Spectroscopic Survey catalog of hard X-ray sources and 12 yr of IceCube muon track data. First, upon performing a stacked search, no significant emission was found. Second, we searched for neutrinos from a list of 43 candidate sources and found an excess from the direction of two sources, the Seyfert galaxies NGC 1068 and NGC 4151. We observed NGC 1068 at flux Φνμ+‾νμ = 4.02^+1.58_-1.52 x 10^-11 TeV⁻¹ cm⁻² s⁻¹ normalized at 1 TeV, with a power-law spectral index γ = 3.10^+0.26_-0.22, consistent with previous IceCube results. The observation of a neutrino excess from the direction of NGC 4151 is at a posttrial significance of 2.9σ. If interpreted as an astrophysical signal, the excess observed from NGC 4151 corresponds to a flux Φνμ+‾νμ = 1.51^+0.99_-0.81 x 10^-11 TeV⁻¹ cm⁻² s⁻¹ normalized at 1 TeV and γ = 2.83^+0.35_-0.28.