A quick search of various databases on the internet indicates that your g-g-grandmother did in fact seek a Civil War pension as a widow for her husband John H. Moberly. He enlisted as a Lieutenant 1st class on 18 June 1861 and was commissioned in Company H, 18th Infantry Regiment, IL (USA)on 30 June 1861. He resigned from this same unit on 10 October 1861. I cannot find any other record of any other service.

In 1890 he filed a pension request (certificate number 1063048). In 1910 his widow Hester Jane Moberly filed a pension request (certificate number 709482).

I'm not sure how you do it, but there is probably a process you can follow to obtain copies of each of these pension request files.