Not really. It isn't normal to write dates in this way. When we write dates we use digits, either '1' or '1st', before or after the month. We say 'first', 'second' and so on, but we don't write these words in dates.

It's not typical, but in formal situations, like a wedding invitation, you will see "October first" or "October twenty-third", etc.

What are you using it for? General terms you can use either but for flyer use October 1 or 1st. In essays it is a little different in what you spell out and what you just write as number- going back to my school days of learning.

Yes you can write it as "October first" but it's most commonly written as "October 1st". You could also say "The first of October".