You have a number of options here.
You have provided limited information so I will mention some available options to you.
Option 1: Quota question data
I have extracted this from the Lighthouse Studio Help: Distinguishing Over-Quota vs. Disqualified - Some researchers need to track whether a respondent was disqualified due to not logically qualifying for the quota versus the respondent qualified for the quota but the quota was already filled. Lighthouse Studio records a quota cell value of -1 if respondents did not qualify due to logic and -2 if respondents qualified but the quota was already filled. You can refer to these values within Skip logic if you need to redirect people to a different terminate screen based on one condition or the other. For example, all respondents are first sent to the same disqualification question as defined on the Settings tab of the Quota dialog. However, you can create "preskips" associated with that page which skip the respondent to the appropriate terminate screen depending on whether the quota cell value is -1 or -2.
Option 2. Disposition codes
When a quota is filled, you can send the respondent to a different terminate question depending on the type of quota fail and for each terminate question you can insert a different disposition code. The disposition code informs you of what type of quota fail the respondent experienced.
Option 3. Using SetValue and a Pass-In Field