I have a collection of Abe Rich cues and some Izzy Rich Cues.
Abe Rich moved to Jacksonville, Florida from Long Island, NY, and continued making Cues, creating "Florida Cues" as his brand with palm trees in the logo.
Izzy Rich moved to Miami, Florida, and his brands were "Rich Q" using the logo you show in these photos, and "Miami Cues".
Correction: Having just read the obituary of Abe Rich, it reveals that after surviving the Nazi's in WWII by making shaving brushes and razor handles for officers serving in Lithuania and toys for their children, and later at the Dachau death camp, he carved wooden clogs for fellow prisoners and wooden canteens for the guards who tormented them, Abe Rich joined his older brother Morris in Miami: "In 1962, Abe decided to join Morris, who was running a successful wood shop in Miami. He stopped in New York to visit an uncle, Izzy Rutschaisky, cue maker to celebrities like Fred Astaire and Erroll Flynn, and stayed a year to apprentice." In Miami, Abe worked with Morris until 1973 then opened his own shop. As Abe Rich of Miami Beach, he created custom cue sticks in a tiny workshop at 428 Jefferson Ave. Initially Florida Cue, the company now is called Star Cue Manufacturing, listed in the Blue Book of Pool Cues. (Interestingly, the logo on my Cue shows an address for Miami Cues of 98 N W 29th St. Miami, FL)
Though he never advertised, Abe sold to professionals as well as amateurs. Minnesota Fats once bought four cues and Jackie Gleason, who portrayed the legendary pool shark in The Hustler, bought one inlaid with diamonds, rubies and sapphires.
Bottom line is that "Rich Q" and "Miami Cue" are brands of Izzy Rich, whereas "Florida Cue" was a brand of Abe Rich.
Attached are 2 photos of examples side by side.