You're probably following up on your query here.
As I've mentioned there - usually when people talk about "qualified dividends", what they mean is "dividends qualified for the IRC Sec. 1(h)(11) treatment". That section defines dividends that qualify for special tax treatment and taxed at capital gains rates.
You'll need to read the section for specifics and exclusions, but in essence it's a dividend paid by a US corporation, or a qualified foreign corporation, in which you've held shares for more than 60 days within a 121 days period around the ex-dividend date.
If I have a dividend paying stock in my portfolio, how would I know whether a dividend is qualified or non-qualified?
You'll need to check the requirements in the IRC Sec. 1(h)(11) and confirm that the dividend qualifies based on these requirements. Usually brokers do that for you and categorize the dividends qualified in 1099-DIV (box 1b).
Do all dividends start as non-qualified dividends and then they become classified as qualified dividends after the 60 day threshold?
Non-qualified dividends are just.... dividends. Sometimes referred to as "ordinary dividends". So yes, all dividends start as non-qualified dividends, and after you satisfy the holding period requirement some may become qualified. Others may become qualified for other treatments (for example in your earlier question you were asking about dividends qualified for Sec. 199A treatment).
Some dividends don't qualify because they're not paid by a qualified entity (see the IRC Sec. 1(h)(11) for definitions and exclusions). For example if the paying entity is not a corporation but rather is something else (trust maybe). Others may not qualify because they're not actually dividends but are rather capital gains distributions, returns of capital, or something else.