Quiero conversar con el médico.

Because Spanish often drops the subject pronoun when it is clear from the verb ending.

• quiero already means I want
• so yo is usually unnecessary

You can say yo quiero, but it adds emphasis, contrast, or clarity. For example:

• Yo quiero conversar con el médico, no con la recepcionista.
  = I want to speak with the doctor, not the receptionist.

In a neutral sentence, Quiero conversar con el médico sounds completely natural.

Quiero is the first person singular present tense of querer.

The infinitive is querer = to want / to love.
Its present-tense forms are irregular:

• yo quiero
• tú quieres
• él/ella/usted quiere
• nosotros queremos
• vosotros queréis
• ellos/ellas/ustedes quieren

So in this sentence, quiero means I want.

Because in Spanish, when one conjugated verb is followed by another verb, the second verb is often left in the infinitive.

So:

• quiero conversar = I want to talk
• necesito hablar = I need to speak
• prefiero esperar = I prefer to wait

Only the first verb, quiero, is conjugated here.
The second verb, conversar, stays in its dictionary form.

All three can relate to talking, but they are not always identical.

• hablar = the most general and common verb for to speak / to talk
• conversar = to converse, often suggesting a more back-and-forth conversation
• charlar = to chat, usually more informal

In many situations, especially in everyday Spain Spanish, hablar con el médico may sound more natural than conversar con el médico.

So:

• Quiero hablar con el médico = very common
• Quiero conversar con el médico = correct, understandable, slightly more formal or less common in some contexts
• Quiero charlar con el médico = more casual, and not always appropriate in a medical context

Because Spanish only makes these two mandatory contractions:

• a + el = al
• de + el = del

But con + el does not contract.

So you say:

• con el médico
• para el médico
• por el médico

not shortened forms.

That is why con el médico is correct.

Spanish uses the definite article more often than English.

Here, el médico means the doctor.
In many real situations, Spanish prefers the article with professions or roles when referring to a specific person.

So:

• Quiero conversar con el médico. = I want to talk with the doctor.

If you said just Quiero conversar con médico, that would be ungrammatical in standard Spanish.

The accent mark shows which syllable is stressed.

• médico is pronounced with stress on the first syllable: MÉ-di-co

Without the accent mark, standard Spanish stress rules would make you expect stress elsewhere, so the written accent is needed.

This is also useful because the stress pattern helps you sound more natural.

In traditional grammar:

• el médico = the male doctor
• la médica = the female doctor

In real usage, la médico is also heard in some places, but in modern standard Spanish, especially in Spain, la médica is very common and widely accepted.

So depending on the doctor, you could say:

• Quiero conversar con el médico.
• Quiero conversar con la médica.

It is grammatical, but by itself quiero can sound a bit direct, depending on context.

If you are speaking to reception staff in a clinic or hospital, Spanish often uses softer phrasing such as:

• Quisiera hablar con el médico. = I would like to speak with the doctor.
• Me gustaría hablar con el médico. = I would like to speak with the doctor.
• ¿Podría hablar con el médico? = Could I speak with the doctor?

So your original sentence is correct, but a learner should know that a softer version may sound more polite in service situations.

Because el médico is not the direct object here.

The structure is:

• Quiero = I want
• conversar = to talk
• con el médico = with the doctor

The doctor is part of a prepositional phrase introduced by con, not a direct object.

The personal a is used before specific human direct objects, for example:

• Veo al médico. = I see the doctor.

But here you are not seeing the doctor as a direct object of the verb. You are talking with him, so you say:

• con el médico

not a el médico.

It is correct and understandable in Spain, but in many everyday situations people are more likely to say:

• Quiero hablar con el médico.

That is probably the most natural version in a clinic, hospital, or reception desk context.

So think of it like this:

• conversar = correct, a bit more formal or less common in this situation
• hablar = the most natural everyday choice

If your goal is very natural Spain Spanish, Quiero hablar con el médico is often the safest option.