No, "does" cannot have two syllables. This is because your jaw only drops once when you say the word. The way I was taught however many times you say a word that is how many syllables there are in that word. Hopes this helps.
The word leasehold has two syllables. The syllables in the word are lease-hold.
there are two syllables in the word grueling
Their are two syllables in the word silly.
The word lady has two syllables. The syllables of the word are la-dy.
the word people has two syllables the word people has two syllables
There are two syllables in the word 'glaciers'.
There are two syllables in bristle
Spir-it has two syllables.
The word noble has two syllables.
There is two syllables in lawyer.
Coughing has two syllables.
Ginger has two syllables.