Mr. McKenzie --
Thanks for your inquiry.
The 1st Florida Infantry Regiment and the 1st Florida Cavalry Regiment were in service of different governments, so there is no connection. A man named Wesley Mathews belonged to Co. "F", 1st Fla. (U.S.) Cavalry, but no others having a similar name in this regiment. Evidently your source doesn't match information available on members of this command at the National Archives.
There are men named William Mathews on roll with the 3rd, 4th and 8th Florida Regiments, plus the 1st Florida (C.S.) Cavalry and the 1st Florida Reserves. Being age 50 in 1863, your William Mathews by law was too old for regular military service. He still could have been enrolled in the Florida state militia from Jackson County. Militiamen were sometimes captured by the Federals, taken off to prison and died while in captivity.
I'm interested in learning more about Godwin's 15th Cavalry. If this is the 15th Confederate Cavalry, I would like to know more about who 'Godwin' might be. Can you provide further information?