Were they in Ufford, Nhants back to 1420? I couldn't be sure before the 1520s, but they might have been in Gresham, Norfolk, back to the 1460s.
Anyway, this is probably a bit better than Geneanet.
First, in Visitations of Northamptonshire, 1564 and 1618-19, there is a pedigree of Quarles of Ufford:
https://archive.org/details/visitationsnort00vincgoog/page/n206/mode/2up
This has John – George – Francis – George and several other children, three of them named John.
No dates are given, but from other evidence, the first John was born before 1500 and maybe as early as 1440. The earliest called “of Ufford” was the first George.
This Common Pleas record, from 1524, has a defendant George Quarles, of Ufford, "Lincolnshire", gentleman, who must be the same person.
http://aalt.law.uh.edu/AALT3/H8/CP40no1042/aCP40no1042fronts/IMG_0450.htm
I couldn’t find any earlier in Ufford, but there was a John Quarles of Gresham, Norfolk, mercer, who occurs several times in Common Pleas in the late 15th century.
For example here, in 1470:
http://aalt.law.uh.edu/AALT2/E4/CP40no837/aCP40no837fronts/IMG_0075.htm
And here, in 1465:
http://aalt.law.uh.edu/AALT2/E4/CP40no814/bCP40no814dorses/IMG_1514.htm
Agnes Qwarles, of Gresham, widow; John Quarles, of Gresham, yeoman, executors of Thomas Quarles, of Gresham, mercer.
If he was an executor of his father's will in 1465, then he must have been born around 1440 and his father around 1400 to 1415.