Here is my opinion:

It may depend on the nature of your research questions and study goal.  If L2 language proficiency is important to the study, then changing the syntax changes the data, and you should not do this.

If the fluency of the participants is not important to your study, you can use brackets to indicate that you have made a correction:  "It was important to know what [he] thought."

If the interview was held in the native language of the participants and translated for analysis and quoting, then you can make minor changes that reflect alternative valid translations without indicating the change.

I'm not sure absolute authenticity is required; instead, I would consider the nature of your goals and how that matches the identities of your participants. In particular, if their non-native status is relevant to the research, then by all means use one of the approaches suggested earlier. But if there is no need to distinguish between native and non-native speakers, then I would recommend making the corrections -- perhaps with a note in the text to say that you did so.

If the purpose of the analysis is only on the content of what participants say, as opposed to a focus on how participants express content, then I don't think transcription of grammatical "mistakes" is necessary. But if you are interested in how participants framed and interpreted what they talked about, then, in my view, close attention to how they said things (grammatical errors and all), exact transcription is important. Hope this helps. 

Thank you all for your replies. In this particular case the non-native status is of no significant relevance for the research, within which the focus lies upon their profession and those two persons have lived in the country and worked in their field for many years.

In Norway, articles that precede a noun are gender-specific, such as "le" and "la" in the french language, and the ending of the nouns are adapted to the article. In their narrations, these articles and the endings are sometimes confused, which would turn bracketing of mistakes and respective corrections into a long and time-consuming process. The alternative to make those corrections and including a remark on it as a note seems to be a good solution to avoid disturbances in the reading flow.

Thank you for your valuable inputs!

Thank you all for your replies. In this particular case the non-native status is of no significant relevance to the research, within which the focus lies upon the profession of the research participants and those two persons have lived in the country and worked in their field for many years.

The above mentioned mistakes relate mainly to the use of articles preceding a noun. In Norway, those are gender-specific, such as "le" and "la" in the french language, and the ending of the nouns are adapted to the article. In the narrations of the participants, these articles and the endings happen to be confused, which would turn bracketing of mistakes and respective corrections into a long and time-consuming process. The alternative to make those corrections and including a remark on it as a note seems to be a good solution to avoid disturbances in the reading flow.

Thank you for your valuable inputs!