Yes, you can. Jira won't care about the trigger, as long as it can read/write the tables as normal.
If you upgrade Jira, you'll need to check and possibly update the trigger.
Although I've just said it's ok to do this, one word of warning about the Jira database - you must NOT write to it. If, as you say, your trigger is going to update things in another database, that's fine, but not the Jira database (or another server's Jira database for that matter)
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