Ownership of an item doesn't convey copyright. The copyright of the Bible would have been to the publishing house when they published that edition. What was written in the family record section technically would be copyrighted to whomever wrote it.
That being said, how you would cite it would be to cite the Bible like any other book if you still have the title page: exact title, publisher, publishing place, date of publication. Note the provenance of the Bible ("Mary Smith Jones was the original owner of this Bible. As of 14 November 2023, it was in the possession of John Jones, 123 Main St., Anywhere, USA.") and anything else that could be useful in identifying it or evaluating it later. ("The writing on the 'Births' page is faded and difficult to read.")