No. DocSearch currently does not support click events.
You can submit your feedback on this GitHub issue: Click and Conversion Analytics · Issue #1045 · algolia/docsearch · GitHub17 and follow the issue to be kept up-to-date on the progress.
The free DocSearch plan, doesn't include technical web support from the Support team. For further assistance, you can post your question on the Algolia Community Forum or Discord. For front-end issues, you can raise the issue directly on GitHub. In addition, you can review our DocSearch Documentation and FAQ.