A code is said to be equidistant if the distance between any two distinct codewords of the code is the same. In this paper, we have studied equidistant single-orbit cyclic and quasi-cyclic subspace codes. The orbit code generated by a subspace $U$ in $\mathbb{F}_{q^n}$ such that the dimension of $U$ over $\mathbb{F}_q$ is $t$ or $n-t$, $\mbox{where}~t=\dim_{\mathbb{F}_q}(\mbox{Stab}(U)\cup\{0\})$, is equidistant and is termed a trivial equidistant orbit code. Using the concept of cyclic difference sets, we have proved that only the trivial equidistant single-orbit cyclic subspace codes exist. Further, we have explored equidistant single-orbit quasi-cyclic subspace codes, focusing specifically on those which are sunflowers.