We consider a special kind of finite element stabilization technique for a singularly perturbed stationary advection-diffusion-reaction boundary value problem. This technique allows us to use a quantum linear systems solver combined with a certain heuristic procedure for estimation of the optimal value of the stabilization parameter. The proposed heuristic procedure is based on a carefully selected loss function that measures the rate of parasitic oscillations. Thus, it gives us the possibility of smoothing the solution by running a simple iterative optimization algorithm against that function. Such an approach leads to a sequence of intermediate finite element solutions, each of which is needed only for loss computation. The special structure of the loss function allows us to compute it as a linear functional of the numerical solution. This property enables the use of a quantum linear solver with a swap test for all those intermediate solutions to estimate the loss for given values of the stabilization parameter. To control the error along with stabilization we add a mesh adaptivity loop with appropriate error estimators constructed. That is, we run an outer adaptation loop, and at each adaptation iteration we employ an inner loop for estimating the stabilization parameter, which can be implemented using quantum linear system solver calls. Computational experiments show that the proposed hybrid algorithm requires fewer classical iterations and smaller meshes compared with typical adaptive schemes. Another useful feature is that it generates non-oscillatory approximations from the very beginning, giving the researcher or engineer a way to explore the qualitative nature of the solution even at the first adaptation steps. Moreover, the constructed algorithm gives us a possible general methodology for developing quantum-classical stabilized finite element schemes.