I've no personal experience, but while researching online I found a study by a university in Bangladesh on the hatch ability of duck eggs through the year. They found that the hatch ability was highest in March, and lowest in June. In June they got the most unfertilised eggs, followed by March, January, April and May.
Another study shows that chicken eggs have the highest hatch ability in in Winter (78.62%), followed by monsoon season (76.70%) and then Summer (75.79%)
They used 3 different breeds for this study, Fayoumi, White Leghorn and Rhode Island Red. Of the 3 the Fayoumi's eggs had the highest hatch rate (78.34%), followed by the White Leghorn (76.48%) and the Rhode Island Red (74.59%)
Age wise they found the hens aged 41-60 weeks produced the most hatchable eggs (77.99%), followed by 61 weeks and older (76.49%) and the 28-40 weeks old (75.71%)
I'll see if I can find anything on Quails...
Same Bangladesh University also did a study on Japanese quail eggs. Results in short:
"Eggs of medium weight, 9.10-10g could be hatched satisfactorily between 4-7 days of pre-incubation holding periods during any season of the year"
I looked a bit more and found another study that said quail eggs should not be stored longer than 9 days prior to incubation at any season of the year. After 18 days the hatch ability of the eggs used during the study "showed no sign of fertility"