You can't differentiate empirically between Bertrand and Cournot oligopoly/duopoly just by looking whether quantity or price varies.
In both Cournot and Bertrand model price can change. The difference is that in Bertrand model causality goes from price to quantity and in Cournot model from quantity to price but that can be hard to disentangle just by eyeballing it.
A textbook example of real-world Cournot oligopoly are OPEC countries (see Mankiw Principles of Economics 8th ed pp 345). OPEC is excellent example because OPEC countries cannot directly "set" world market price of oil (at least not in a sense that grocery can put price tags on its goods). Instead
OPEC tries to set production levels for each of the member countries. The problem that OPEC faces is much the same as problem that Jack and Jill face in our story [referring to previous example of Cournot model].
Even outside textbooks OPEC is typically modeled using quantity competition models (e.g. see Lawell 2020).
However, you should note that in equilibrium where there would be no perturbations (no new oil discovered, no change in preference etc) oil prices would appear to be fixed. Conversely, the retail prices change even if not a often as commodity prices.
By the same token OPEC quantity produced varies over time. Companies/countries have to always respond to new perturbations to factors that form supply and demand (e.g. costs of production, preferences etc).
You shouldn't just assume that because prices tend to be less volatile market is competing on prices instead of quantity.