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Is it correct to ask : How did it go at doctors ? Not sure how to ask

There are two aspects to this: using the article, and how to punctuate.

(1) Using the article

In English, apart from a few exceptions, all common nouns like doctor require a "determiner", which includes articles (a/an and the), demonstratives (such as this and that), possessive pronouns (e.g. her, your) and quantifiers (e.g. few or all).

In your sentence, it's presumed that the reference is to a particular doctor, so the definite article ("the") would be typical. It would also be common to refer to that doctor or your doctor.

(2) Do we need an apostrophe?

This is a bit more complex, as it depends on whether the question is about your experience with the doctor or at the doctor's rooms/clinic, and also whether you're referring to a single doctor or multiple doctors.

If you use "at", then the question is about the place. The word "rooms" or "clinic" is assumed, and doesn't need to be spoken or written. Nonetheless, if you're writing the sentence it's important to indicate the possessive by using the apostrophe: doctor's. You would write:

How did it go at the doctor's?

[In the unusual situation that the person asking the question knew you were seeing more than one doctor, you would put the apostrophe after the plural form: doctors'].

However, it's quite standard to use "with" instead of "at". In this case, the reference is to the person rather than the location, so there's no possessive. You would write:

How did it go with the doctor?

This would also be the more appropriate usage if the doctor visited you rather than you going to them. The plural would also be more common in this situation, where a series of different doctors/surgeons had seen you (for example at your hospital bed):

How did it go with the doctors?

In spoken English

The difficulty when you hear the word "doctors" is that it might not be apparent whether the speaker means the possessive (doctor's), the plural (doctors) or the possessive plural (doctors'). Fluent speakers learn from experience to use a different construction if the meaning wouldn't be clear from the context - hence we would probably choose "with" if there were multiple doctors. Nonetheless, ambiguity between plural and possessive is not uncommon, and offers a rich vein for comedians and sitcoms.

The expression is not correct. It would properly be "How did it go at the doctor's." You need the 'the' preceding 'doctor's' or else which doctor they went to is ambiguous. It is necessary to say 'doctor's' rather than 'doctors' because the implied object is 'the doctor's office,' with office being omitted.

How did it go at the doctor's?