I have done a simple calculation for $G$-equivariant quantum K-ring for $\mathcal{O}(-n)$ bundle over $\mathbb{CP}^N$ by using 3d $\mathcal{N}=2$ $U(1)$ gauge theory with $N+1$ flavor of charge 1 and one flavor of charge $-n$ whose Higgs branch is . The result is as follows:
\begin{equation}
QK^\bullet_G(\mathcal{O}(-n)\to \mathbb{CP}^N)\simeq \mathbb{C}\left[p^{\pm1},\tau^{\pm},\eta^{\pm},\mu_i^{\pm}\right]/\mathcal{I}\,,\quad i=1,\dots, N+1\,,
\end{equation}
where the ideal $\mathcal{I}$ is given by
\begin{equation}
\prod_{i=1}^{N+1}(p-\mu_i)=\tau (p-\eta)^n~.
\end{equation}
Note that $\mu_i$ are the equivariant parameters of $GL(N+1,\mathbb{C})$ and $\eta$ is that of $\mathbb{C}^*$.
I have also computed the K-theoretic Givental $J$-function:
\begin{equation}
J[\mathcal{O}(-n)\to \mathbb{CP}^N]=\sum_{k\ge0}\tau^k\frac{(q^n\eta;q)_{nk}}{\prod_{j=1}^{N+1}(q\mu_j/\mu_i;q)_k}~.
\end{equation}