Not in the UI that I can think of directly.
But the log will have "indexing" lines in it, and I think the admin audit logging will catch it as well, as long as it is enabled.
I'm not entirely sure how much use it might be though - the index is also updated whenever an issue is touched, so technically, the system was partially re-indexed the last time you used it! I'm guessing you mean one of the full re-indexes though (or project re-index?)
I think re-index events are not included in Audit Log (System --> Audit Log), or at least I don't see them and Atlassian documentation don't mention it (https://confluence.atlassian.com/adminjiraserver/auditing-in-jira-applications-938847740.html).
It would be great that this info was included there: re-index type (project or global background/blocking), start and finish.
Meanwhile, I prefer to take a look into log files. It's clear and easy to know when re-index started and finished.
It appears to me that the Issues folder is timestamped with the current date, however, the minimum date of the files within the folder does seem to be the date of the last index. I specifically validated against a background re-index.
It's unclear if the timestamps can be used to definitely identify start/end times of the re-index, but the date appears correct.
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