Does Month Take an S?

If there is no noun that the time expression is connected to then months acts like a regular noun and you just need to add an s to make it plural. I need to start work in one month.

Does months need an apostrophe?

In simple terms: "X months' + noun (or noun phrase, etc.)" requires an apostrophe as it means "[the amount of of] X months' [worth] of noun" (e.g. Three months' wages = three months [worth] of wages = the value of the wages for three months), the genitive apostrophe is required.

Is it 3 month or 3 months?

It should be : within a three-month period(note the hyphen) OR simply within three months(no period, no apostrophe).