Short answer: yes, indirectly. Lemon juice will not switch cortisol off, but the vitamin C in lemons supports the body's stress response, and staying properly hydrated prevents the physical stress that triggers hormone spikes in the first place.
Why lemon shows up in every 'cortisol cocktail'
Lemon juice is the base of the popular adrenal drinks doing the rounds online because it pairs vitamin C with fluid and a pleasant sour hit that makes an electrolyte mix drinkable. Seedless lemons are simply easier to juice at volume — the effect is the same as any fresh lemon.
The adrenal-supporting recipe
Juice of 1 lemon · ½ cup orange juice · 1 cup coconut water (natural electrolytes) · ¼ teaspoon sea salt (sodium to support the adrenals) · 200 mg magnesium powder (promotes relaxation). Mix thoroughly, top up with sparkling water to taste, and sip through the day rather than downing it at once.
The forest-produce version
Our Netrang women's unit makes the same idea from what grows locally: amla instead of imported citrus concentrate, tulsi and lemongrass for the aromatics, and unrefined rock salt. Amla carries far more vitamin C per gram than lemon, and it is a Minor Forest Produce the community already collects.
A sensible caveat
No drink replaces sleep, movement and lower workload. If your cortisol pattern is genuinely disordered, see a physician. Treat this as hydration you actually enjoy, not medicine.