Not really it technically means the same thing, just like saying the amount of words also, but I will say in English we don't use the phrase the quantity of words that much we'd use the number or the amount more.

Hello there, thank you La Liseuse and Nanren888 for the explanations. I really appreciate your help. :)

As Nanren888 has pointed out, it is grammatically wrong to use 'amount' with a countable noun. You will hear many native speakers saying such things as 'amount of people' or 'amount of times', but it this is - strictly speaking - incorrect. Please do not say 'amount of words', as you may have this marked as incorrect if you were to write it in a formal exam, for example.

Never amount of words. Never amount of countable objects. Quantity is usually also used for uncountable. A quantity of salt.... An amount of salt... A quantity of water.... An amount of water.... A number of words.... A number of people.... A number of people will incorrectly use "amount" with countable, plural nouns. The amount of writing with this sort of thing in it is increasing. :(

I moved my response from comments to the answer area.