No, GDPR does not apply here — but other laws might .

The General Data Protection Regulation only concerns itself with the following:

This Regulation applies to the processing of personal data wholly or partly by automated means and to the processing other than by automated means of personal data which form part of a filing system or are intended to form part of a filing system.

per Article 2.

Furthermore, there is a specific exemption (Art. 2 (2)(c)) which would seem to apply to you (emphasis mine):

This Regulation does not apply to the processing of personal data:

(a) in the course of an activity which falls outside the scope of Union law;

(b) by the Member States when carrying out activities which fall within the scope of Chapter 2 of Title V of the TEU;

(c) by a natural person in the course of a purely personal or household activity;

(d) by competent authorities for the purposes of the prevention, investigation, detection or prosecution of criminal offences or the execution of criminal penalties, including the safeguarding against and the prevention of threats to public security.

You are sharing the conversation in a purely personal capacity and are not processing it wholly or partly through automated means nor are you intending it to form part of a filing system.

Additionally, depending on the contents of the conversation and what you share, there may not be any personal data involved at all. For example, if you only share the messages themselves (and they don't contain any personal data that could identify someone) and censor/black out any names or photographs/avatars from the senders. Either way, GDPR does not appear to apply to you in this context.

However, it appears that Article 18.1 of the Spanish Constitution grants the right to "personal and family privacy", as does the Organic Act 1/982 on Civil Protection of the Right to Honour, Personal and Family Privacy, and Self-Image.

It's possible that, depending on the contents of the conversation, the other party involved in the conversation may have a case for action against you in the civil or criminal courts (for example, if the information invades their privacy or is defamatory in some way) if you share the contents of the conversation with someone else. I am not familiar with Spanish law so cannot comment further on this matter.