We introduce univariate bicycle (UB) codes, a structured subclass of generalized bicycle (GB) quantum low-density parity-check (LDPC) codes obtained via a Frobenius relation. This construction reduces the code design space from a two-polynomial search in GB codes to a single-polynomial search, while preserving sparsity. We provide an explicit algebraic characterization of the logical coset spaces by constructing a basis for the logical quotient space, yielding a complete parametrization of logical operators. Leveraging this structure, we derive upper bounds on the minimum distance by relating structured logical representatives to cycle-density properties of associated circulant matrices. Finally, simulation results for short- to medium-length UB codes (block lengths ranging from a few hundred to approximately $10^3$) demonstrate competitive performance relative to existing GB and bivariate bicycle (BB) codes despite the additional algebraic restriction.