Immanants, interpreted in a broad sense, are certain polynomials in commuting variables whose coefficients are functions from the symmetric group S_n to the complex numbers. Functions most often used in this context are S_n-characters. Goulden and Jackson expressed irreducible character immanants as coefficients in generating functions given by permanents and determinants. Merris and Watkins gave similar expressions for induced character immanants. Deforming the commutative coordinate ring in n2 variables into the noncommutative quantum coordinate ring, and S_n into the noncommutative Hecke algebra H_n(q), we define quantum immanants for irreducible and induced H_n(q) characters. We show that these arise as coefficients in natural quantizations of the formulae given by Goulden-Jackson and Merris-Watkins. Moreover, our quantized Goulden-Jackson formula is related to the quantized Master Theorem of Garoufalidis-Le-Zeilberger in much the same way that Goulden-Jackson's original formula is related to MacMahon's classical Master Theorem.