This paper proposes a high-speed hardware accelerator for QR-UOV, a multivariate scheme, that executes all three operations: key generation, signature generation, and signature verification. QR-UOV utilizes a quotient polynomial ring structure to reduce the public-key size of the original UOV scheme; however, this introduces functional requirements distinct from other multivariate schemes, such as polynomial-matrix operations over Fqℓ\mathbb{F}_{q^\ell}, coefficient expansion for the Mersenne prime field Fq\mathbb{F}_{q}, and the expansion of symmetric matrices generated in a compressed form. To accelerate polynomial-matrix multiplication over Fqℓ\mathbb{F}_{q^\ell}, the proposed architecture employs a D×DD \times D systolic array (DD is the parallelization factor), in which each PE internally contains a polynomial multiply-accumulate unit. For signature verification, the architecture also incorporates a quadratic-form operation unit without explicitly storing the intermediate vector--matrix products. In addition, to ensure a regular data supply from key expansion to matrix operations, the architecture incorporates on-the-fly rejection sampling, which exploits the property that AES-CTR output blocks can be generated independently, as well as an expansion process for the symmetric matrices in compressed form. Furthermore, the architecture enables pipelined processing, overlapping the ii-th polynomial-matrix operation with the (i+1)(i+1)-th public-key expansion. FPGA evaluation shows that the proposed hardware executes key generation, signature generation, and signature verification in 0.74 ms, 0.28 ms, and 0.19 ms, respectively, at security level I. Furthermore, we confirm that the proposed hardware exhibits a lower LUT-time product than that of existing UOV hardware.