The verb 'have' has three main functions:
1. To indicate possession of some kind. For example: 'I have a laptop'. This corresponds to the Spanish 'tener'.
There are two possible negative forms when 'have' has this meaning: either
'I haven't got a laptop'
'I don't have a laptop'.
The first is more informal, but both are correct.
2. As an auxiliary verb for perfect tenses. For example 'I have been to London'.
This corresponds to the Spanish 'haber'.
The negative form of the auxiliary is haven't or hasn't.
'I haven't been to London'.
3. To replace other common verbs, such as 'take', 'hold', 'eat'. For example 'I have a shower every morning' 'I have dinner with my family.' In Spanish you would use the more specific verbs.
The negative of have in this situation is 'don't/doesn't/didn't have'.
'I don't have dinner with my family'.
As the verb 'have' is representing an ordinary verb (such as 'take') in these sentences it has to conjugate like a normal verb ie with 'do' as its auxiliary.
In your sentence above, 'She has interesting conversations' is an example of the 3rd type of construction. This is why you have to form the negative as 'doesn't have'. The other two negative forms cannot be used in this situation.
I hope that helps.