Nearly all quinolone antibiotics in use are fluoroquinolones, which contain a fluorine atom in their chemical structure and are effective against both Gram-negative and Gram-positive bacteria like ciprofloxacin, Moxi/Levofloxacin one of the most widely used antibiotics worldwide.

Quinolones or 4-quinolones, are synthetic carboxylic acid derivatives. Various alterations of the four quinolone ring structures have produced numerous broad-spectrum antimicrobial agents. Fluoroquinolone compounds became available for use in the mid1980s. Fluoroquinolones were created by substitutions of a fluorine moiety at position 6. Although the non-fluorinated quinolones only feature a moderately extended Gram-negative spectrum, fluoroquinolones were developed for activity against both Gram-negative and Gram-positive organisms, chlamydiae and mycoplasmas. They are effective against some penicillin non-susceptible or multi-drug resistant pneumococci and some methicillin resistant Staphylococcus aureus (MRSA).

Microsoft Word - fluoroquinolones.DOC (la.gov)

The quinolone antimicrobial agents that work against DNA gyrase are more effective against gram-negative bacteria. Fluoroquinolone antibiotics are a newer derivative of quinolones, and they are capable of antibacterial activity against both topoisomerase IV and DNA gyrase simultaneously.

I guess fluoroquinolone is a group name, and it means a group of antibiotic. Quinolone is a chemical structure, there are different quinolone structures available, but not all are drugs.

Reference:

Title: Handbook of Antibiotics

Publisher: Lippincott Williams & Wilkins

Year of publication: 2000

copyright: Richard E Reese and Robert F Betts

Quinolones and fluoroquinolones differ in their activity against the two enzymes produced by bacteria, topoisomerase IV and DNA gyrase.

https://www.drugs.com/drug-class/quinolones.html

The quinolones can be differentiated within classes based on their pharmacokinetic properties. The newer fluoroquinolones have broad-spectrum bactericidal activity, excellent oral bioavailability, good tissue penetration and favorable safety and tolerability profiles.

https://www.aafp.org/pubs/afp/issues/2000/0501/p2741.html

Quinolone are basically type of antibiotic, its inhibits the growth of bacteria while fluoroquinolones are another class of antibiotic and thay are fluiroquinolones because their structures are derived from floride