It does not appear that the two regiments served together but I'm still checking.
The 6th Ill. Cavalry Rgt. was in Grierson's Raid from LaGrange, Tenn. to Baton Rouge, La., April 17-May 2, 1863--considered one of the most spectacular Union cavalry raids of the war. Raid helped divert Confederate attention away from Grant's final (and successful) move against Vicksburg.
Dee Brown wrote a book called Grierson's Raid which is well done and the movie "The Horse Soldiers" with John Wayne is loosely based on Grierson's raid.
I do not find a history of the 6th Ill. Cavalry
For the 5th Illinois Cavalry, see 4th Iowa Cavalry Regiment, "The Story of a Cavalry Regiment: The Career of the Fourth Iowa Veteran Volunteers from Kansas to Georgia, 1861-1865," by William Forse Scott, Adjutant, 4th Iowa Volunteers. Originally released in 1893 by G. P. Putnam's Sons, New York, New York, with 602 pages, this excellent regimental can also be read as a history of the 3rd Iowa, the 5th and 11th Illinois, and the 10th Missouri (US) cavalry regiments, as they were brigaded with the 4th. "The best history of a Hawkeye cavalry unit; a detailed narrative, maps, full index recount the regiment's campaigns in the Western theater," (reviewed by Allan Nevins, for Civil War Books). The 4th Iowa Cavalry fought at Vicksburg, Memphis, Columbus, and Brice's Crossroads against Nathan Bedford Forrest's Confederate Cavalry. An original copy of the 1893 release may be hard to locate, includes the original introduction by Oscar H. La Grange of the 1st Wisconsin Cavalry 630 pages, maps, 1992 reprint by Camp Pope Bookshop, costs $40