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.
Their are two syllables in the word silly.
there are two syllables in the word grueling
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 but two syllables in 'lurid'.
There is two syllables in lawyer.
There are two syllables in bristle
There are two syllables in the word 'glaciers'.
Spir-it has two syllables.
The word noble has two syllables.
Puppet has two syllables.