**No**
At typical dosages, Menthol's side effects do not include drowsiness. They do include:
> Severe allergic reactions (rash; hives; itching; difficulty breathing; tightness in the chest; swelling of the mouth, face, lips, or tongue); redness or irritation at the application site. [source]
At overdose levels, Menthol's side-effects do not include drowsiness. They do include:
> * Bladder and kidneys
> * Blood in the urine
> * No urine output
> * Lungs
> * Rapid breathing
> * Shallow breathing
> * Gastrointestinal
> * Abdominal pain
> * Diarrhea
> * Nausea
> * Vomiting
> * Heart and blood
> * Rapid heartbeat
> * Nervous system
> * Convulsions
> * Dizziness
> * Tremor
> * Unconsciousness
> * Unsteady walking
>
>
> [source]
Use medications according to the instructions. On average, they are more reliable than the Internet.