In the Q1 model, this paper proposes a low-data quantum key-recovery attack against Lai-Massey structures, Misty structures, Type-1 generalized Feistel structures, SMS4-like generalized Feistel structures and MARS-like generalized Feistel structures. This attack only needs to select constant-sized plain-ciphertexts, analyze the encryption process of block cipher structures, and recover the key by searching and calculating some intermediate states and round keys using Grover’s algorithm. This attack belongs to the Q1 model, which is more practical than the Q2 model since no quantum superposition query is required. For the 3-round Lai-Massey structure, compared with other quantum attacks, this attack requires only $ {\rm O}(1) $ data and belongs to the Q1 model, and is even reduced by the $ n{2^{n/4}} $ factor on the evaluation of the complexity product (time×data×classical memory×quantum bits). For the 6-round Misty structure, this attack still retains the advantage of low data complexity, and especially for the 6-round Misty L/R-FK structure, this attack is reduced by the$ {2^{n/2}} $factor on the evaluation of the complexity product. For the 9-round 3-branch Type-1 generalized Feistel structure, in line with other quantum attacks on the evaluation of the complexity product, this attack still retains the advantage of low data complexity and belongs to the chosen plaintext attack. In addition, a low-data quantum key-recovery attack for SMS4-like generalized Feistel structures and MARS-like generalized Feistel structures are also given in this study, complementing their security evaluation in the Q1 model.