We present a family of non-CSS quantum stabilizer codes using the structure of duadic constacyclic codes over $\mathbb{F}_4$. Within this family, quantum codes can possess varying dimensions, and their minimum distances are bounded by a square root bound. For each fixed dimension, this allows us to construct an infinite sequence of binary quantum codes with a growing minimum distance. Additionally, we demonstrate that this quantum family includes an infinite subclass of degenerate codes with the mentioned properties. We also introduce a technique for extending splittings of duadic constacyclic codes, providing new insights into the minimum distance and minimum odd-like weight of specific duadic constacyclic codes. Finally, we establish that many best-known quantum codes belong to this family and provide numerical examples of quantum codes with short lengths within this family.

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