There are many ways to produce queens.This is a our way, including grafting, cell building and cell introduction into mating nucs.
The absolute best time for producing queen cells is during swarming season. That's when colonies are really ready to make cells and is usually when they're in their peak condition. That being said you can make them anytime the drone population and weather is good enough for mating.


Incubator:
http://www.gqfmfg.com/12-volt-hova-ba...
https://www.amazon.com/1588-Genesis-H...
Grafting tool:
https://www.mannlakeltd.com/chinese-g...
Cell protectors;
https://www.mannlakeltd.com/top-bar-c...
Cell cups:
https://www.mannlakeltd.com/push-in-c...
Feeder:
https://www.mannlakeltd.com/pro-feede...

References:
Standard methods for artificial rearing of Apis mellifera larvae
https://www.researchgate.net/publicat...

THE QUALITY OF HONEY BEE QUEENS FROM QUEEN CELLS INCUBATED AT DIFFERENT TEMPERATURES
https://www.researchgate.net/publicat...

An Introduction to Queen Honey Bee Development, Penn State Ext.
https://extension.psu.edu/an-introduc... development